diff --git "a/9tA0T4oBgHgl3EQfO_-M/content/tmp_files/load_file.txt" "b/9tA0T4oBgHgl3EQfO_-M/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/9tA0T4oBgHgl3EQfO_-M/content/tmp_files/load_file.txt" @@ -0,0 +1,1906 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf,len=1905 +page_content='Computational analysis of NM-polynomial based topological indices and graph-entropies of carbon nanotube Y-junctions Sohan Lal1, Vijay Kumar Bhat1,∗, Sahil Sharma1 1School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, Jammu and Kashmir, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' sohan1993sharma@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='com, vijaykumarbhat2000@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='com, sahilsharma2634@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='com Abstract Carbon nanotube Y-junctions are of great interest to the next generation of innovative multi-terminal nanodevices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Topological indices are graph-theoretically based parameters that describe various structural properties of a chemical molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The entropy of a graph is a topological descriptor that serves to characterize the complexity of the underlying molecular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The concept of entropy is a physical property of a thermodynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Graph entropies are the essential thermophysical quantities defined for various graph invariants and are applied to measure the heterogeneity and relative stabilities of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In this paper, several neighborhood degree sum-based topological indices including graph-based entropies of carbon nanotube Y-junction graphs are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Keywords: Armchair carbon nanotube, graph entropy, NM-polynomial, topological indices, Y- junction graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' MSC (2020): 05C10, 05C35, 05C90 1 Introduction Nanotechnology is currently popular because of its evolving, electron transfer property and low-cost implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Nanotubes [1], were discovered in 1985 and carbon nanotubes [2] in 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In nanoscience and technology, branched or non-straight carbon nanotubes such as L, T, X, and Y have a lot of applications in electronic devices, such as three-terminal transistors, multi-terminal nanoelectronics, switches, amplifiers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', [3, 4, 5, 6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' These junctions are a great option for the production of nanoscale electronic devices with better switching and reliable transport properties at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For more applications of carbon nanotube Y-junctions, we refer to [9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The first proposed branched carbon nanotube was of Y shape, commonly known as Y-junction or three-terminal junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' These junctions are classified as an armchair, zig-zag, or chiral depending on the chirality of connected carbon nanotubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Also, they can be single-walled or multi-walled, symmetric or asymmetric, capped or uncapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' A carbon nanotube is called uncapped if both ends are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' A Y-junction is called symmetric if the nanotubes joining in the Y shape are identical, heptagons appeared isolated, and are distributed symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For various symmetric and asym- metric carbon nanotube Y-junctions, we refer to [12, 13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' A carbon nanotube Y-junction is formed by joining three identical carbon nanotubes in a Y- shaped pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' These junctions contain exactly six hexagons as well as heptagons at the branch- ing points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The first structural model of symmetrical single-walled armchair carbon nanotube Y- junctions was proposed by Chernozatonskii [16] and Scuseria [17], independently, in 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' These junctions were experimentally observed [18] in 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For more applications and properties of carbon nanotube Y-junction graphs, we refer to [19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Mathematical chemistry is a branch of theoretical chemistry that employs mathematical tech- niques to explain the molecular structure of a chemical molecule and its physicochemical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Molecular graphs are a visual representation of a chemical molecule with vertices representing atoms and edges representing bonds between the atoms [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let G = (V (G), E(G)) be a molecular graph with vertex set V (G) and edge set E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The order of a molecular graph G is defined as the total number of vertices in G, denoted by |V (G)|, and the number of edges in G is called size of G, denoted by |E(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Any edge of the graph connecting its vertices u and v, is denoted by e = uv ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Two vertices of graph G are said to be adjacent if there exists an edge between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The degree of vertex v ∈ V (G), denoted by d(v), is defined as the number of vertices that are adjacent to 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='02169v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='mes-hall] 3 Jan 2023 vertex v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', d(v)= |{u : e = uv ∈ E(G)}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The neighborhood degree sum of vertex v ∈ V (G) is denoted by dn(v), and is defined as the sum of the degrees of all vertices that are adjacent to v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', dn(v) = � u d(v): uv ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The minimum cardinality of the set K ⊆ V (G) such that G \\ K is disconnected graph is called connectivity or vertex-connectivity of a connected graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The connected graph G is said to be k-connected if its connectivity is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Topological indices are the numerical values calculated from molecular graphs to describe various structural properties of the chemical molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' They are frequently used to model many physico- chemical properties in various quantitative structure-property/activity relationship (QSPR/QSAR) studies [23, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In 1947, the chemist Harold Wiener [26] initiated the concept of topological indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Since then, various topological indices have been introduced, and a lot of research has been conducted toward computing the indices for different molecular graphs and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' A topological index based on the degree of end vertices of an edge can predict various physicochemical properties of the molecule, such as heat of formation, strain energy, entropy, enthalpy, boiling points, flash point, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', without using any weight lab [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The Zagreb indices and their variations have been used to investigate molecular complexity, ZE- isomerism, and chirality [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In general, the Zagreb indices have shown applicability for deriving multilinear regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Ghorbani and Hosseinzadeh [28] introduced the third version of the Zagreb index and shows that this index shows a good correlation with acentric factor and entropy of the octane isomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [29] introduced neighborhood degree sum-based topological indices namely neighborhood version of forgotten topological index and neighborhood version of second modified Zagreb index and discuss some mathematical properties and degeneracy of these novel indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For more neighborhood degree sum-based topological indices, their properties, and applications, we refer to [24, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The process of computing the topological indices of a molecular graph from their definitions is complex and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Thus, for a particular family of graphs and networks, algebraic poly- nomials play an important role in reducing the computational time and complexity when computing its topological indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In short, with the help of algebraic polynomials, one can easily compute various kinds of graph indices within a short span of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The NM-polynomial plays vital role in the computation of neighborhood degree sum-based topological indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let dn(v) denotes the neigh- borhood degrees sum of vertex v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then, the neighborhood M-polynomial (NM-polynomial) of G is defined as [30, 32, 33] NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = � i≤j |Eij(G)|xiyj (1) where, |Eij(G)|, i, j ≥ 1, be the number of all edges e = uv ∈ E(G) such that {dn(u) = i, dn(v) = j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Recently, various neighborhood degree sum-based topological indices have been computed via the NM-polynomial technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For example, Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [30, 34] obtained some neighborhood and multiplicative neighborhood degree sum-based indices of molecular graphs by using their NM- polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Kirmani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [24] and Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [35], investigated some neighborhood degree sum-based topological indices of antiviral drugs used for the treatment of COVID-19 via the NM- polynomial technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Shanmukha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [36] computed the topological indices of porous graphene via NM-polynomial method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For more neighborhood degree sum-based topological indices via NM- polynomials, we refer to [24, 35, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Some neighborhood degree sum-based topological indices and their derivation from NM-polynomial are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In chemical graph theory, the determination of the structural information content [39] of a graph is mostly based on the vertex partition of a graph to obtain a probability distribution of its vertex set [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Based on such a probability distribution, the entropy of a graph can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Thus, the structural information content of a graph is defined as the entropy of the underlying graph topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The concept of graph entropy or entropy of graph was first time appeared in [41], where molecular graphs are used to study the information content of an organism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Entropy-based methods are powerful tools to investigate various problems in cybernetics, mathematical chemistry, pattern recognition, and computational physics [22, 39, 42, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 2 Table 1: Description of some topological indices and its derivation from NM-polynomial Topological index Formula Derivation from NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) Third version of Zagreb index [28]: NM1(G) � uv∈E(G) � dn(u) + dn(v) � (Dx + Dy)(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Neighborhood second Zagreb index [29]: NM2(G) � uv∈E(G) � dn(u)dn(v) � (DxDy)(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Neighborhood second modified Zagreb index [30]: nmM2(G) � uv∈E(G) � 1 dn(u)dn(v) � (SxSy)(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Neighborhood forgotten topological index [29]: NF (G) � uv∈E(G) � d2 n(u) + d2 n(v) � (D2 x + D2 y)(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Third NDe index [30]: ND3(G) � uv∈E(G) dn(u)dn(v)(dn(u) + dn(v)) DxDy(Dx + Dy)(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Neighborhood general Randic index [30]: NRα(G) � uv∈E(G) dα n(u)dα n(v) (Dα x Dα y )(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Neighborhood inverse Randic index [30]: NRRα(G) � uv∈E(G) 1 dα n(u)dα n(v) (Sα x Sα y )(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Fifth NDe index [30]: ND5(G) � uv∈E(G) � d2 n(u)+d2 n(v) dn(u)dn(v) � (DxSy + SxDy)(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Neighborhood harmonic index [30]: NH(G) � ab∈E(G) 2 dn(u)+dn(v) 2SxT (NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Neighborhood inverse sum indeg index [30]: NI(G) � uv∈E(G) � dn(u)dn(v) dn(u)+dn(v) � (SxT DxDy)(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 where, Dx = x � (∂(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='x,y)) ∂x � , Dy = y � (∂(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='x,y)) ∂y � , Sx = � x 0 NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='t,y) t dt, Sy = � y 0 NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='x,t) t dt, T (NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y)) = NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Entropy is a measure of randomness, uncertainty, heterogeneity, or lack of information in a sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Based on information indices, there are various approaches to deriving graph entropy from the topological structure of a given chemical molecule [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For example, Trucco [39] and Rashevsky [41] defined graph entropies in terms of degree of vertex, extended degree sequences, and number of vertices of a molecular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Tan and Wu [46] study network heterogeneity by using vertex-degree based entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Mowshowitz defined the entropy of a graph in terms of equivalence relations de- fined on the vertex set of a graph and discussed some properties related to structural information [47, 48, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Recently, Shabbir and Nadeem [51] defined graph entropies in terms of topological indices for the molecular graphs of carbon nanotube Y-junctions and developed the regression models between the graph entropies and topological indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Nadeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [52] calculated some degree-based topological indices for armchair carbon semicapped and capped nanotubes and investigated their chemical and physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Baˇca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [53] computed some degree-based topological indices of a carbon nanotube network and studied its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Azeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [54] calculated some M-polynomials based topological indices of carbon nanotube Y-junctions and their variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Ahmad [55], studied some ve-degree based topological indices of carbon nanotube Y-junctions and discussed their proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Ayesha [56] calculated the bond energy of symmetrical single-walled armchair carbon nanotube Y-junctions and developed regression models between bond energy and topological indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Rahul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [57] calculated some degree-based topological indices and graph-entropies of graphene, graphyne, and graphdiyne by using Shannon’s approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The above-mentioned literature and applications of carbon nanotubes in the field of nanoscience and technology inspired us to develop more research on the molecular structure of carbon nanotube Y-junction and their variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In addition, no work has been reported on NM-polynomial based topological indices and index-entropies of Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Therefore, the main contribution of this study includes the following: Computation of NM-polynomials of carbon nanotube Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Computation of some neighborhood degree sum-based topological indices from NM-polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Some graph index-entropies in terms of topological indices are defined and computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 3 Comparative analysis of obtained topological indices and graph index-entropies of Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 2 Aim and Methodology We use the edge partition technique, graph-theoretical tools, combinatorial computation, and the degree counting method to derive our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The degree of end vertices is used to generate the patterns of edge partitions of the Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Using such partitions, a general expression of NM-polynomials is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then, several neighborhood degree sum-based topological indices are obtained from the expression of these NM-polynomials with the help of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Also, graph index- entropies in terms of topological indices have been defined by using edge-weight functions and computed for Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The paper is structured as follows: In Section 3, we define topological index-based graph en- tropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The Y-junction graphs and their constructions are described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In Section 5, the general expression of the NM-polynomials and neighborhood degree sum-based topological indices of Y-junction graphs are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Section 6 describes the graph index-entropies of Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The numerical analysis of the findings is discussed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Finally, the conclusion is drawn and discussed in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 3 Definitions and Preliminaries In this section, we define graph index-entropies in terms of an edge-weight function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In 2008, Dehmer [40] defined the entropy for a connected graph G as follows: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [40] Let G = (V (G), E(G)) be a connected graph of order n and g be an arbitrary information functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then the entropy of G is defined as Hg(G) = − n � i=1 g(vi) n� i=1 g(vi) log � g(vi) n� i=1 g(vi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (2) Since an information function defined on the vertex set of a graph is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Hence, Dehmer’s definition shows the possibility of producing various graph entropies for a variation in the selection of information functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For such graph entropy, we can refer to [58, 59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let β : E(G) → R+ ∪ {0} be an edge-weight function and dn(u) = � uv∈E(G) d(u), denotes the sum of degrees of end vertices of an edges incident to vertex u ∈ V (G) (also known as neighborhood degree-sum of vertex u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' for eight different edge-weight functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' the third-version of Zagreb index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' neighborhood second Zagreb index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' neighborhood forgotten topological index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' neighborhood second modified Zagreb index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' third NDe index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' fifth NDe index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' neighborhood harmonic index and neighborhood inverse sum indeg index-entropies have been defined in the following manner: Third-version of Zagreb index-entropy: If e = uv is an edge of a connected graph G and β1(e) = dn(u) + dn(v) is an edge-weight function defined on E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then, the third-version of Zagreb index is NM1(G) = � e=uv∈E(G) β1(e) = � e=uv∈E(G) dn(u) + dn(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (3) Equation (2) for this edge-weight function gives us Hβ1(G) = − � e∈E(G) β1(e) � e∈E(G) β1(e)log � β1(e) � e∈E(G) β1(e) � = − 1 � e∈E(G) β1(e) � e∈E(G) β1(e) � log(β1(e)) − log � e∈E(G) β1(e) � 4 = − 1 � e∈E(G) β1(e) � e∈E(G) β1(e)log(β1(e)) + 1 � e∈E(G) β1(e) � e∈E(G) β1(e)log � � e∈E(G) β1(e) � = log � � e∈E(G) β1(e) � − 1 � e∈E(G) β1(e) � e∈E(G) β1(e)log(β1(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' On replacing � e∈E(G) β1(e) by NM1(G) in the above equation, we get the following third-version of Zagreb index-entropy Hβ1(G) = log(NM1(G)) − 1 NM1(G) � e∈E(G) β1(e)logβ1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (4) Similarly, we define other graph index-entropies as follows: Neighborhood second Zagreb index-entropy: For β2(e) = dn(u)dn(v), the neighborhood second Zagreb index and neighborhood second Zagreb index-entropy are NM2(G) = � e=uv∈E(G) dn(u)dn(v), (5) and Hβ2(G) = log(NM2(G)) − 1 NM2(G) � e∈E(G) β2(e)logβ2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (6) Neighborhood forgotten topological index-entropy: For β3(e) = d2 n(u) + d2 n(v), the neighborhood forgotten topological index and neighborhood forgotten topological index-entropy are NF(G) = � e=uv∈E(G) d2 n(u) + d2 n(v), (7) and Hβ3(G) = log(NF(G)) − 1 NF(G) � e∈E(G) β3(e)logβ3(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (8) Neighborhood second modified Zagreb index-entropy: For β4(e) = 1 dn(u)dn(v), the neighborhood second modified Zagreb index and neighborhood second modified Zagreb index- entropy are nmM2(G) = � e=uv∈E(G) 1 dn(u)dn(v), (9) and Hβ4(G) = log(nmM2(G)) − 1 nmM2(G) � e∈E(G) β4(e)logβ4(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (10) Third NDe index-entropy: For β5(e) = dn(u)dn(v) � dn(u) + dn(v) � , the third NDe index and third NDe index-entropy are ND3(G) = � e=uv∈E(G) dn(u)dn(v) � dn(u) + dn(v) � , (11) and Hβ5(G) = log(ND3(G)) − 1 ND3(G) � e∈E(G) β5(e)logβ5(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (12) Fifth NDe index-entropy: For β6(e) = dn(u) dn(v) + dn(v) dn(u), the fifth NDe index and fifth NDe index-entropy are ND5(G) = � e=uv∈E(G) dn(u) dn(v) + dn(v) dn(u), (13) and Hβ6(G) = log(ND5(G)) − 1 ND5(G) � e∈E(G) β6(e)logβ6(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (14) 5 Neighborhood harmonic index-entropy: For β7(e) = 2 dn(u)+dn(v), the neighborhood har- monic index and neighborhood harmonic index-entropy are NH(G) = � e=uv∈E(G) 2 dn(u) + dn(v), (15) and Hβ7(G) = log(NH(G)) − 1 NH(G) � e∈E(G) β7(e)logβ7(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (16) Neighborhood inverse sum indeg index-entropy: For β8(e) = dn(u)dn(v) dn(u)+dn(v), the neighbor- hood inverse sum index and neighborhood inverse sum index-entropy are NI(G) = � e=uv∈E(G) dn(u)dn(v) dn(u) + dn(v), (17) and Hβ8(G) = log(NI(G)) − 1 NI(G) � e∈E(G) β8(e)logβ8(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (18) 4 Y-Junction Graphs The Y-junctions examined in this study are created by the covalent connection of three identical single-walled carbon nanotubes crossing at an angle of 120◦ and are uniquely determined by their chiral vector v = nv1 +nv2, where v1 and v2 are graphene sheet lattice vectors and n is non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let m ≥ 1 and n ≥ 4 be an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then, an uncapped symmetrical single-walled carbon nanotube Y-junction is made up of an armchair Y (n, n) and three identical single-walled armchair carbon nanotubes Tm(n, n) each of length m (layers of hexogones), denoted by Y m(n, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In Y m(n, n), we have 3 4n2 − 3 2n + 5 faces including three openings (where the tubes meet to the amchair) each of chirality (n, n), six heptagones, and 3 4n2 − 3 2n − 4 hexagones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In addition, the tube Tm(n, n) contains 2mn hexagonal faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let n, m, and l be positive integers with m ≥ 1 and n = 2l, for some l ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then J = Jm(n, n) be the Y -junction graph of Y m(n, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' It has 9l2 − 3l + 2 hexagonal rings along with six heptagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The graph J is of order 6l2 +18l +6+24ml and size 9l2 +21l +9+36ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' It has 6l2 +12l +6+24ml vertices of degree three and 12l vertices of degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Note that graph J is a 2-conneced graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Along with 2-connected Y-junction graph J, the 1-connected Y-junction graphs have also been taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' These graphs are obtained by adding pendants to the degree 2 vertices of the 2-connected graph J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Note that, each tube of J has 2n vertices of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Therefore, the graph J has 6n vertices of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The graph obtained by connecting 2n pendants to any one tube in J is denoted by J1, and we call it as second type Y-junction graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The order and size of graph J1 are 6l2 + 22l + 6 + 24ml and 9l2 + 25l + 9 + 36ml, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The graph J2 represents a graph which is obtained by attaching 4n pendants to any two tubes of J and we call it as third type Y-junction graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In J2, we have 6l2 + 26l + 6 + 24ml vertices and 9l2 + 29l + 9 + 36ml edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The graph obtained by joining 6n pendants to all the three tubes of J is denoted by J3, and we called it as fourth type Y-junction graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' It has 6l2 + 30l + 6 + 24ml vertices and 9l2 + 33l + 9 + 36ml edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The carbon nanotube Y-junction graphs J, J1, J2, and J3 are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The edge partition of Y-junction graphs J, J1, J2, and J3 based on the neighborhood degree-sum of end vertices of an edge is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 6 (a) Y-junction graph J (b) Y-junction graph J1 (c) Y-junction graph J2 (d) Y-junction graph J3 Figure 1: A symmetrical uncapped single-walled armchair carbon nanotubes Y-junction graphs Table 2: Edge partitions of J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' J1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' and J3 dn(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) J-frequency J1-frequency J2-frequency J3-frequency (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) 0 4l 8l 12l (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) 6l 4l 2l 0 (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) 12l 8l 4l 0 (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) 0 2l 4l 6l (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) 0 4l 8l 12l (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) 6l 4l 2l 0 (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) 12l 8l 4l 0 (9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) 9l2 − 15l + 36ml + 9 9l2 − 9l + 36ml + 9 9l2 − 3l + 36ml + 9 9l2 + 3l + 36ml + 9 5 NM-Polynomials and Topological Indices of Y-Junction Graphs In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' we develop the general expression of NM-polynomials for the Y-junction graphs and then recover various neighborhood degree-sum based topological indices from these polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J be the Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then NM(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = 6lx5y5 + 12lx5y8 + 6lx8y8 + 12lx8y9 + (9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotubes has 9l2 +21l +9+36ml number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let E(i,j) be the set of all edges with neighborhood degree sum of end vertices i, j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', E(i,j) = {uv ∈ E(J) : dn(u) = i, dn(v) = j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 7 Extension of J to JiBy means of structural analysis of J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' the edge set of J can be partitioned into five sets on the basis of neighborhood degree sum of end vertices as follows: E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) = {uv ∈ E(J) : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 5},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(J) : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(Jm(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' n)) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J) : dn(u) = 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' and |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5)| = 6l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 12l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 6l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 12l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 9l2 − 15l + 9 + 36ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Equation (1), the NM-polynomial of J is obtained as follows: NM(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = � i≤j |E(i,j)|xiyj = |E(5,5)|x5y5 + |E(5,8)|x5y8 + |E(8,8)|x8y8 + |E(8,9)|x8y9 + |E(9,9)|x9y9 = 6lx5y5 + 12lx5y8 + 6lx8y8 + 12lx8y9 + (9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J be the Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then (i) NM1(J) = 162l2 + 246l + 648ml + 162 (ii) NM2(J) = 729l2 + 663l + 2916ml + 729 (iii) NF(J) = 1458l2 + 1446l + 5832ml + 1458 (iv) nmM2(J) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 (v) NRα(J) = 6l(25α + 2(40)α + 64α + 2(72)α) + 81α(9l2 − 15l + 9 + 36ml) (vi) ND3(J) = 13122l2 + 6702l + 52488ml + 13122 (vii) ND5(J) = 18l2 + 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='86l + 72ml + 18 (viii) NH(J) = l2 + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='69l + 4ml + 1 (ix) NI(J) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 (x) S(J) = 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7l2 + 714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='23l + 4670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9ml + 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let f(x, y) = NM(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = 6lx5y5 +12lx5y8 +6lx8y8 +12lx8y9 +(9l2 −15l+9+36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then, we have Dx(f(x, y)) = 30lx5y5 + 60lx5y8 + 48lx8y8 + 96lx8y9 + 9(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Dy(f(x, y)) = 30lx5y5 + 96lx5y8 + 48lx8y8 + 108lx8y9 + 9(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' D2 x(f(x, y)) = 150lx5y5 + 300lx5y8 + 384lx8y8 + 768lx8y9 + 81(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' D2 y(f(x, y)) = 150lx5y5 + 768lx5y8 + 384lx8y8 + 972lx8y9 + 81(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' DxDy(f(x, y)) = 150lx5y5 + 480lx5y8 + 384lx8y8 + 864lx8y9 + 81(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (Dx + Dy)f(x, y) = 60lx5y5 + 156lx5y8 + 96lx8y8 + 204lx8y9 + 18(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' DxDy(Dx + Dy)f(x, y) = 1500lx5y5 + 6240lx5y8 + 6144lx8y8 + 14688lx8y9 + 1458(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (D2 x + D2 y)f(x, y) = 300lx5y5 + 1068lx5y8 + 768lx8y8 + 1740lx8y9 + 162(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Dα xDα y (f(x, y)) = 6l(25)αx5y5 + 12l(40)αx5y8 + 6l(64)αx8y8 + 12l(72)αx8y9 + (81)α (9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 8 SxSy(f(x, y)) = 6l 25x5y5 + 12l 40 x5y8 + 6l 64x8y8 + 12l 72 x8y9 + (9l2−15l+9+36ml) 81 x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' SyDx + SxDy(f(x, y)) = 12lx5y5 + 267l 10 x5y8 + 12lx8y8 + 145l 6 x8y9 + 2(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 2SxT(f(x, y)) = 6l 5 x10 + 24l 13 x13 + 3l 4 x16 + 24l 17 x17 + (9l2−15l+9+36ml) 9 x18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' SxTDxDy(f(x, y)) = 15lx10 + 480l 13 x13 + 384l 16 x16 + 864l 17 x17 + 81(9l2−15l+9+36ml) 18 x18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' S3 xQ−2TD3 xD3 y(f(x, y)) = 93750l 512 x8+ 768000l 1331 x11+ 1572864l 2744 x14+ 4478976l 3375 x15+ 531441(9l2−15l+9+36ml) 4096 x16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Now, using Table 1 we have (i) NM1(J) = (Dx + Dy)f(x, y)|x=y=1 = 162l2 + 246l + 648ml + 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (ii) NM2(J) = (DxDy)f(x, y)|x=y=1 = 729l2 + 663l + 2916ml + 729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (iii) NF(J) = (D2 x + D2 y)f(x, y)|x=y=1 = 1458l2 + 1446l + 5832ml + 1458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (iv) nmM2(J) = (SxSy)f(x, y)|x=y=1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (v) NRα(J) = (Dα xDα y )f(x, y)|x=y=1 = 6l(25α+2(40)α+64α+2(72)α)+81α(9l2−15l+9+36ml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (vi) ND3(J) = DxDy(Dx + Dy)f(x, y)|x=y=1 = 13122l2 + 6702l + 52488ml + 13122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (vii) ND5(J) = SyDx + SxDy(f(x, y))|x=y=1 = 18l2 + 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='86l + 72ml + 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (viii) NH(J) = 2SxT(f(x, y))|x=y=1 = l2 + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='69l + 4ml + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (ix) NI(J) = SxTDxDy(f(x, y))|x=y=1 = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (x) S(J) = S3 xQ−2TD3 xD3 y(f(x, y))|x=y=1 = 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7l2 + 714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='23l + 4670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9ml + 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J1 be the second type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then NM(J1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = 4lx3y7+4lx5y5+8lx5y8+2lx7y7+4lx7y9+4lx8y8+8lx8y9+(9l2−9l+9+36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The second type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotubes has 9l2 +25l+9+36ml edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let E(i,j) be the set of all edges with neighborhood degree sum of end vertices i, j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', E(i,j) = {uv ∈ E(J1) : dn(u) = i, dn(v) = j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' By means of structure analysis of J1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' the edge set of J1 can be partitioned into eight sets on the basis of neighborhood degree sum of end vertices as follows: E(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) = {uv ∈ E(J1) : dn(u) = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) = {uv ∈ E(J1) : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 5},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(J1) : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) = {uv ∈ E(J1) : dn(u) = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J1) : dn(u) = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(J1) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J1) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J1) : dn(u) = 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' and |E(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 8l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7)| = 2l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 8l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 9l2 − 9l + 9 + 36ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Equation (1), the NM-polynomial of J1 is obtained as follows: NM(J1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = � i≤j |E(i,j)|xiyj = |E(3,7)|x3y7 + |E(5,5)|x5y5 + |E(5,8)|x5y8 + |E(7,7)|x7y7 + |E(7,9)|x7y9 + |E(8,8)|x8y8 + |E(8,9)|x8y9 + |E(9,9)|x9y9 = 4lx3y7 + 4lx5y5 + 8lx5y8 + 2lx7y7 + 4lx7y9 + 4lx8y8 + 8lx8y9 + (9l2 − 9l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J1 be the second type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then 9 (i) NM1(J1) = 162l2 + 314l + 648ml + 162 (ii) NM2(J1) = 729l2 + 957l + 2916ml + 729 (iii) NF(J1) = 1458l2 + 2074l + 5832ml + 1458 (iv) nmM2(J1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 (v) NRα(J1) = 2l(2(21)α + 2(25)α + 4(40)α + (49)α + 2(63)α + 2(64)α + 4(72)α) + (81)α(9l2 − 9l + 9 +36ml) (vi) ND3(J1) = 13122l2 + 12170l + 52488ml + 13122 (vii) ND5(J1) = 18l2 + 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='328l + 72ml + 18 (viii) NH(J1) = l2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98l + 4ml + 1 (ix) NI(J1) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='15l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 (x) S(J1) = 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7l2 + 1178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92l + 4670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9ml + 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Refer to Theorem 2 for proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J2 be the third type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then NM(J2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = 8lx3y7+2lx5y5+4lx5y8+4lx7y7+8lx7y9+2lx8y8+4lx8y9+(9l2−3l+9+36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The third type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotubes has 9l2 + 29l + 9 + 36ml number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let E(i,j) be the set of all edges with neigh- borhood degree sum of end vertices i, j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', E(i,j) = {uv ∈ E(J2) : dn(u) = i, dn(v) = j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' By means of structure analysis of J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' the edge set of J2 can be partitioned into eight sets on the basis of neighborhood degree sum of end vertices as follows: E(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) = {uv ∈ E(J2) : dn(u) = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) = {uv ∈ E(J2) : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 5},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(J) 2 : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) = {uv ∈ E(J2) : dn(u) = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J2) : dn(u) = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(J2) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J2) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J2) : dn(u) = 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' and |E(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7)| = 8l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5)| = 2l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 8l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 2l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 9l2 − 3l + 9 + 36ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Equation (1), the NM-polynomial of J2 is obtained as follows: NM(J2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = � i≤j |E(i,j)|xiyj = |E(3,7)|x3y7 + |E(5,5)|x5y5 + |E(5,8)|x5y8 + |E(7,7)|x7y7 + |E(7,9)|x7y9 + |E(8,8)|x8y8 + |E(8,9)|x8y9 + |E(9,9)|x9y9 = 8lx3y7 + 2lx5y5 + 4lx5y8 + 4lx7y7 + 8lx7y9 + 2lx8y8 + 4lx8y9 + (9l2 − 3l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J2 be the third type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then (i) NM1(J2) = 162l2 + 382l + 648ml + 162 (ii) NM2(J2) = 729l2 + 1251l + 2916ml + 729 (iii) NF(J2) = 1458l2 + 2478l + 5832ml + 1458 (iv) nmM2(J2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='819l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 10 (v) NRα(J2) = 2l(4(21)α+(25)α+2(40)α+2(49)α+4(63)α+(64)α+2(72)α)+(81)α(9l2−3l+9+36ml) (vi) ND3(J2) = 13122l2 + 17638l + 52488ml + 13122 (vii) ND5(J2) = 18l2 + 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='56l + 72ml + 18 (viii) NH(J2) = l2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='57l + 4ml + 1 (ix) NI(J2) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='048l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 (x) S(J2) = 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7l2 + 1643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='61l + 4670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9ml + 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Refer to Theorem 2 for proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J3 be the fourth type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then NM(J3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = 12lx3y7 + 6lx7y7 + 12lx7y9 + (9l2 + 3l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The fourth type Y-junction graph of an uncapped symmetrical single-walled armchair car- bon nanotube has 9l2 + 33l + 9 + 36ml number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let E(i,j) be the set of all edges with neighborhood degree sum of end vertices i, j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', E(i,j) = {uv ∈ E(J3) : dn(u) = i, dn(v) = j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' By means of structure analysis of J3, the edge set of J3 can be partitioned into four sets on the basis of neighborhood degree sum of end vertices as follows: E(3,7) = {uv ∈ E(J3) : dn(u) = 3, dn(v) = 7}, E(7,7) = {uv ∈ E(J3) : dn(u) = 7, dn(v) = 7}, E(7,9) = {uv ∈ E(J3) : dn(u) = 7, dn(v) = 9}, E(9,9) = {uv ∈ E(J3) : dn(u) = 9, dn(v) = 9}, and |E(3,7)| = 12l, |E(7,7)| = 6l, |E(7,9)| = 12l, |E(9,9)| = 9l2 + 3l + 9 + 36ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Equation (1), the NM-polynomial of J3 is obtained as follows: NM(J3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = � i≤j |E(i,j)|xiyj = |E(3,7)|x3y7 + |E(7,7)|x7y7 + |E(7,9)|x7y9 + |E(9,9)|x9y9 = 12lx3y7 + 6lx7y7 + 12lx7y9 + (9l2 + 3l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J3 be the fourth type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then (i) NM1(J3) = 162l2 + 450l + 648ml + 162 (ii) NM2(J3) = 729l2 + 1545l + 2916ml + 729 (iii) NF(J3) = 1458l2 + 3330l + 5832ml + 1458 (iv) nmM2(J3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 (v) NRα(J3) = 6l(2(21)α + (49)α + 2(63)α) + (81)α(9l2 + 3l + 9 + 36ml) (vi) ND3(J3) = 13122l2 + 23106l + 52488ml + 13122 (vii) ND5(J3) = 18l2 + 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='90l + 72ml + 18 (viii) NH(J3) = l2 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='090l + 4ml + 1 (ix) NI(J3) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 (x) S(J3) = 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7l2 + 2085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l + 4670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9ml + 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Refer to Theorem 2 for proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 11 6 Graph Index-Entropies of Y-Junction Graphs In this section, we compute the index-entropy of carbon nanotube Y-junctions in terms of neigh- borhood degree sum-based topological indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' We first compute index-entropies of the Y-junction graph J whose edge partition is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Third-version of Zagreb index-entropy of J From part (i) of Theorem 2, we have NM1(J) = 162l2 + 246l + 648ml + 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (19) Now, from Equation (4), the third-version of Zagreb index-entropy of J is Hβ1(J) = log(NM1(J)) − 1 NM1(J) � e∈E(J) β1(e)logβ1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (20) Using Table 2 and Equation (19) in Equation (20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' we get the required third-version of Zagreb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='index-entropy of J as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='Hβ1(J) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='log(NM1(J)) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='NM1(J) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e∈E(J) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='β1(e)logβ1(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='log(162l2 + 246l + 648ml + 162) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='162l2 + 246l + 648ml + 162 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6l(10)(log10) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='12l(13)(log13) + 6l(16)(log16) + 12l(17)(log17) + (9l2 − 15l + 36ml + 9)(18)(log18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='log(162l2 + 246l + 648ml + 162) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='162l2 + 246l + 648ml + 162 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='60l(log10) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='156l(log13) + 96l(log16) + 204l(log17) + (162l2 − 270l + 648ml + 162)(log18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='log(162l2 + 246l + 648ml + 162) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='162l2 + 246l + 648ml + 162 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='60l(1) + 156l(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1139433523) +96l(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2041199827) + 204l(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2304489214) + (162l2 − 270l + 648ml + 162)(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2552725051) � ≈ log(162l2 + 246l + 648ml + 162) − 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='78l + 810ml + 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 162l2 + 246l + 648ml + 162 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Neighborhood second Zagreb index-entropy of J From part (ii) of Theorem 2, we have NM2(J) = 729l2 + 663l + 2916ml + 729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (21) By using the values given in Table 2 and Equation (21) in Equation (6), we get the required neigh- borhood second Zagreb index-entropy of J as follows: Hβ2(J) = log(NM2(J)) − 1 NM2(J) � e∈E(J) β2(e)logβ2(e) = log(729l2 + 663l + 2916ml + 729) − 1 729l2 + 663l + 2916ml + 729 � 6l(25)(log25) + 12l(40)(log40) + 6l(64)(log64) + 12l(72)(log72) + (9l2 − 15l + 36ml + 9)(81)(log81) � ≈ log(729l2 + 663l + 2916ml + 729) − 1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22l2 + 958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='27l + 5564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88ml + 1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22 729l2 + 663l + 2916ml + 729 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Similarly, we compute the remaning index-entropies of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Table 3 shows some calculated graph index-entropies of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In this way, the topological index-based entropies for Y-junction graphs J1, J2, and J3 are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The index-based entropies of J1 J2, and J3 are given in Tables 4, 5, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 12 Table 3: Index-entropies of J Entropy Values of entropies Hβ3(J) log(1458l2 + 1446l + 5832ml + 1458) − 3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l2+2601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l+12885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84ml+3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 1458l2+1446l+5832ml+1458 Hβ4(J) log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='207l2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='828ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 Hβ5(J) log(13122l2 + 12170l + 52488ml + 13122) − 41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75l2+15202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='13l+166059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31ml+41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75 13122l2+12170l+52488ml+13122 Hβ6(J) log(18l2 + 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='86l + 72ml + 18) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41l2+14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='81l+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='67ml+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41 18l2+44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='86l+72ml+18 Hβ7(J) log(l2 + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='69l + 4ml + 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l2+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72l+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='81ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95 l2+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='69l+4ml+1 Hβ8(J) log(40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) − 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='45l2+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='18l+105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='82ml+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='45 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2+59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24l+162ml+40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Table 4: Index-entropies of J1 Entropy Values of entropies Hβ1(J1) log(162l2 + 314l + 648ml + 162) − 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31l2+346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='09l+813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24ml+203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 162l2+314l+648ml+162 Hβ2(J1) log(729l2 + 957l + 2916ml + 729) − 1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22l2+1523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='54l+5564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88ml+1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22 729l2+957l+2916ml+729 Hβ3(J1) log(1458l2 + 2074l + 5832ml + 1458) − 3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46l2+3991l+12885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84ml+3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 1458l2+2074l+5832ml+1458 Hβ4(J1) log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='207l2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='065l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='897ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 Hβ5(J1) log(13122l2 + 12170l + 52488ml + 13122) − 41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75l2+32699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='53l+166059ml+41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75 13122l2+12170l+52488ml+13122 Hβ6(J1) log(18l2 + 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='328l + 72ml + 18) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41l2+19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='09l+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='67ml+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41 18l2+56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='328l+72ml+18 Hβ7(J1) log(l2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98l + 4ml + 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9l2+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='21l+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 l2+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98l+4ml+1 Hβ8(J1) log(40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='15l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) − 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37l2+36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84l+105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='48ml+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2+75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='15l+162ml+40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Table 5: Index-entropies of J2 Entropy Values of entropies Hβ1(J2) log(162l2 + 382l + 648ml + 162) − 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31l2+430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='65l+813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24ml+203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 162l2+382l+648ml+162 Hβ2(J2) log(729l2 + 1251l + 2916ml + 729) − 1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22l2+2088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='77l+5564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88ml+1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22 729l2+1251l+2916ml+729 Hβ3(J2) log(1458l2 + 2478l + 5832ml + 1458) − 3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46l2+5380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37l+12885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84ml+3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 1458l2+2478l+5832ml+1458 Hβ4(J2) log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='819l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='099l2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='007l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='396ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='819l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 Hβ5(J2) log(13122l2 + 17638l + 52488ml + 13122) − 41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75l2+50196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l+166059ml+50196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95 13122l2+17638l+52488ml+13122 Hβ6(J2) log(18l2 + 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='56l + 72ml + 18) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41l2+23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44l+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='67ml+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41 18l2+65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='56l+72ml+18 Hβ7(J2) log(l2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='57l + 4ml + 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9l2+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='61l+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 l2+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='57l+4ml+1 Hβ8(J2) log(40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='048l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) − 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37l2+46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='387l+105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='48ml+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2+91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='048l+162ml+40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 13 Table 6: Index-entropies of J3 Entropy Values of entropies Hβ1(J3) log(162l2 + 450l + 648ml + 162) − 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31l2+515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='23l+813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24ml+203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 162l2+450l+648ml+162 Hβ2(J3) log(729l2 + 1545l + 2916ml + 729) − 1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22l2+2654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='14l+5564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88ml+1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22 729l2+1545l+2916ml+729 Hβ3(J3) log(1458l2 + 3330l + 5832ml + 1458) − 3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46l2+6769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75l+12885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84ml+3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 1458l2+3330l+5832ml+1458 Hβ4(J3) log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='18l2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='35l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 Hβ5(J3) log(13122l2 + 23106l + 52488ml + 13122) − 41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75l2+67694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4l+166059ml+41514 13122l2+23106l+52488ml+13122 Hβ6(J3) log(18l2 + 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='90l + 72ml + 18) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41l2+27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='79l+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='67ml+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41 18l2+75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='90l+72ml+18 Hβ7(J3) log(l2 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='090l + 4ml + 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9l2+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='04l+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 l2+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='090l+4ml+1 Hβ8(J3) log(40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) − 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37l2+56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44l+105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='48ml+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2+106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l+162ml+40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 7 Numerical Results and Discussions The numerical values of topological indices and graph index-entropies of Y-junction graphs are com- puted in this section for some values of l and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In addition, we plot line and bar graphs for comparison of the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Here, we use the logarithm of the base 10 for calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The numerical values of topological indices for Y-junction graph J are given in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The logarithmic values of Table 7 are plotted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From the vertical axis of Figure 2, we can conclude that for Y-junction graph J, the topological indices have the following order: nmM2 ≤ NR−1/2 ≤ NH ≤ ND5 ≤ NI ≤ NM1 ≤ NM2 ≤ S ≤ NF ≤ ND3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The third NDe index has the most dominating nature compared to other topological indices, whereas neighborhood second modified Zagreb index grew slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Table 7: Numerical values of topological indices for Y-junction graph J [l, m] NM1(J) NM2(J) NF (J) NmM2(J) NR− 1 2 (J) ND3(J) ND5(J) NH(J) NI(J) S(J) [2,2] 3894 16635 33510 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='55 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7436 288966 467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='38 968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92 25950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='56 [3,3] 8190 35523 71406 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='27693 623718 962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='58 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='07 2040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='63 55857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='79 [4,4] 14106 61701 123882 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='39 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='81026 1089690 1637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='76 3517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='34 97442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22 [5,5] 21642 95169 190938 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='96 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='3436 1686882 2492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='3 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='45 5399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='05 150703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 [6,6] 30798 135927 272574 23.' metadata={'source': 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243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4103 3274926 4742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='02 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='83 10377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='47 292258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 [8,8] 53970 239313 479586 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='27 317.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4769 5387850 7711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='74 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='21 16975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='89 480522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 [10,10] 83622 371859 744918 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0103 6641142 9466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 20882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 592170 Figure 2: Graphical comparison among topological indices of Y-junction graph J 14 indices 1000000 NM,(J) 100000 NF(J) nmr M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='(J) NR 10000 D (J 1000 NH(J) NI(J) S(J) 100 10 [4,4] [5,5] [6,6] [7,7] [2,2] [3,3] [8,8] [9,9] [10,10] [1,m]Table 8 shows some numerical values of topological indices for Y-junction graph J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The logarith- mic values of these topological indices are plotted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Figure 3, we can conclude that the topological indices for Y-junction graph J1 have the following order: nmM2 ≤ NH ≤ NR−1/2 ≤ ND5 ≤ NI ≤ NM1 ≤ NM2 ≤ S ≤ NF ≤ ND3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Also, we see that the logarithemic values of NR−1/2 and NH for J1 are almost same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Table 8: Numerical values of topological indices for Y-junction graph J1 [l, m] NM1(J1) NM2(J1) NF (J1) nmM2(J1) NR− 1 2 (J1) ND3(J1) ND5(J1) NH(J1) NI(J1) S(J1) [2,2] 4030 17223 35166 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='34052 299902 490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='656 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='96 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 26879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='94 [3,3] 8394 36405 74190 7.' metadata={'source': 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97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='68103 1111562 1683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='312 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92 3581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 99300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98 [5,5] 21982 96639 196578 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 146.' metadata={'source': 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2448102 3595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='968 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88 7781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 218430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 [7,7] 42050 186033 378086 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1918 3313202 4822.' metadata={'source': 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+page_content='624 352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84 13601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 384269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9] 68598 304587 618714 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='14 443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5323 5437062 7814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='952 441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='82 17119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='35 484704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 [10,10] 84302 374799 761198 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7026 6695822 9581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='28 540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 21042 596816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 Figure 3: Graphical comparison among topological indices of Y-junction graph J1 Table 9 shows some calculated values of topological indices for Y-junction graph J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The log- arithmic values of these indices are plotted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The vertical axis of Figure 4 shows the comparison clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Figure 4 shows that the logarithmic values of ND3 are extremely high when compared to other topological indices of J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Figure 4, we see that the graph of NR−1/2 and NH are almost coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Table 9: Numerical values of topological indices for Y-junction graph J2 [l, m] NM1(J2) NM2(J2) NF (J2) nmM2(J2) NR− 1 2 (J2) ND3(J2) ND5(J2) NH(J2) NI(J2) S(J2) [2,2] 4166 17811 35574 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='948 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='49121 310838 509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='12 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='14 1032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='596 27809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='32 [3,3] 8598 37287 74502 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='517 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='23681 656526 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='68 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='71 2136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='144 58645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='93 [4,4] 14650 64053 128010 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='186 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98241 1133434 1720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='28 3644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='692 101159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 [5,5] 22322 98109 196098 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='955 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='728 1741562 2595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='85 5558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24 155350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 [6,6] 31614 139455 278766 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='824 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2192 2480910 3651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='36 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='42 7876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='788 221219 [7,7] 42526 188091 376014 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='793 279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2192 3351478 4886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='99 10600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='34 298764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 [8,8] 55058 244017 487842 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='862 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9648 4353266 6302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='48 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='56 13728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88 387987 [9,9] 69210 307233 614250 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='031 448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7104 5486274 7898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='48 447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='13 17262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='43 488886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 [10,10] 84982 377739 755238 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='3 548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='456 6750502 9673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 21200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98 601463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 15 indices 1000000 NM,(J,) 100000 NM,(J,) NF(J)) 10000 NR (J1) ND,(J,) 1000 ND,(J) NH(J,) 100 NI(J) 10 [2,2] [3,3] [4,4] [5,5] [6,6] [7,7] [8,8] [9,9][10,10] [1,m]Figure 4: Graphical comparison among topological indices of Y-junction J2 Table 10 shows some numerical values of topological indices of Y-junction J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Figure 5 depicts the graphical comparison of these indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Table 10 and Figure 5 show that the values of topological indices strictly increase as the values of l and m increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Tables 7, 8, 9, and 10, we see that as the values of l and m in Y-junction graphs increases, the corresponding values of topological indices grew very fastly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Table 10: Numerical values of topological indices of Y-junction graph J3 [l, m] NM1(J3) NM2(J3) NF (J3) NmM2(J3) NR− 1 2 (J3) ND3(J3) ND5(J3) NH(J3) NI(J3) S(J3) [2,2] 4302 18399 37278 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='15 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6419 321774 529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='18 1064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 28694 [3,3] 8802 38169 77058 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='82 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9628 672930 1055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='27 2183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='85 59973 [4,4] 14922 65229 131418 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='59 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='284 1155306 1761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='36 3708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='3 102929 [5,5] 22662 99579 200358 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='605 1768902 2647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='45 5637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75 157562 [6,6] 32022 141219 283878 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='43 212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='926 2513718 3713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='54 7972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2 223873 [7,7] 43002 190149 381978 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='247 3389754 4959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='3 281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='63 10711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 301861 [8,8] 55602 246369 494658 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='67 363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='568 4397010 6385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72 13856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 391526 [9,9] 69822 309879 621918 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='94 453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='889 5535486 7991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='81 17405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 492868 [10,10] 85662 380679 763758 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='209 6805182 9777 551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 21360 605887 Figure 5: Graphical comparison among topological indices of Y-junction graph J3 A few values of graph index-entropies of Y-junction graph J are listed in Table 11 and illustrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Figure 6, we see that entropy measures of Hβ1, Hβ2, Hβ3, and Hβ8 almost 16 indices 1000000 100000 NF(J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 10000 NR 1000 ND,(J) ND,(J2) 100 NH(J,) NI(J2) 10 [2,2] [3,3] [4,4] [5,5] [6,6] [7,7] [8,8] [9,9][10,10] [1,m]indices 1000000 100000 NM(J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' NM,(J3) NF(J3) 10000 "M,(J3) NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1/2(J3) ND,(J3) 1000 ND,(J3) NH(J3) NI(J3) 100 S(J3) 10 [2,2] [4,4] [5,5] [6,6] [7,7] [3,3] [8,8] [9,9][10,10] [1,m]coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Table 11: Numerical values of index-entropies of J [l, m] Hβ1 (J) Hβ2 (J) Hβ3 (J) Hβ4 (J) Hβ5 (J) Hβ6 (J) Hβ7 (J) Hβ8 (J) [2,2] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='363878 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='349537 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='351098 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='293045 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='469266 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='751708 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='162725 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='567487 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='674998 [4,4] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='912369 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='901799 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} 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3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='303596 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='371357 [8,8] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='488993 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='480834 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='481247 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='437785 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} 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+page_content='581037 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='538669 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='610178 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='065343 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='526328 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='583289 [10,10] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='677788 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='670239 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='670554 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='629107 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='696846 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='154321 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='61983 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='672599 Figure 6: Graphical comparison among index-entropies of J The values of index-entropy of Y-junction graph J1 is listed in Table 12 and illustrated in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Table 12 and Figure 7, we find that measures of graph index-entropies Hβ1, Hβ2, Hβ3, Hβ5, Hβ6, and Hβ1 are almost same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Table 12: Numerical values of index-entropies of J1 [l, m] Hβ1 (J1) Hβ2 (J1) Hβ3 (J1) Hβ4 (J1) Hβ5 (J1) Hβ6 (J1) Hβ7 (J1) Hβ8 (J1) [2,2] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='374116 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='362875 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 [2,2] [3,3] [4,4] [5,5] [6,6] [7,7] [8,8] [9,9] [10,10] [1,m]Figure 7: Graphical comparison among index-entropies of J1 Table 13 depicts some graph index-entropies of Y-junction graph J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The graphical comparison of index-entropies of Y-junction graph J2 is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Figure 8, we see that graph index-entropies of J2 increases as the values of l and m increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Table 13: Numerical values of index-entropies of J2 [l, m] Hβ1 (J2) Hβ2 (J2) Hβ3 (J2) 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='659833 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='744042 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='672602 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='680861 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='628548 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='673185 Figure 8: Graphical comparison among index-entropies of J2 In Table 14, we calculate some graph index-entropies of Y-junction graph J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Figure 9 shows the graphical comparison among index-entropies of J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Table 14 and Figure 9, we see that index entropies Hβ1, Hβ2, Hβ3, Hβ6, and Hβ8 of J3 are almost same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Also, Tables 11, 12, 13, and 14 shows that graph index-entropies of Y-junction graph increases as the values of l and m increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 Hβ(J,) Hβ,(J,) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Hβ,(J,) Hβ(J,) Hβ,(J) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 Hβ,(J,) Hβ,(J,) Hβ(J,) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Index-entropies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 [3,3] [2,2] [4,4] [5,5] [7,7] [6,6] [8,8] [9,9] [10,10] [1,m] Hβ,(J2) Hβ,(J2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Hβ,(J2) I Hβ,(J2) I Hβ,(J2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 Hβ,(J2) Hβ,(J2) Hβ,(J2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Index-entropies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 [3,3] [2,2] [4,4] [5,5] [7,7] [6,6] [8,8] [9,9] [10,10] [1,m] Table 14: Numerical values of index-entropies of J3 [l, m] Hβ1 (J3) Hβ2 (J3) Hβ3 (J3) Hβ4 (J3) Hβ5 (J3) Hβ6 (J3) Hβ7 (J3) Hβ8 (J3) [2,2] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='401692 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='544179 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5964 [10,10] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='683569 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='679896 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='681147 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='420469 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='530087 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='684252 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='632058 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='684668 Figure 9: Graphical comparison among index-entropies of J3 8 Conclusion and Future work In this study, the general expression of NM-polynomial for carbon nanotube Y-junction graphs is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Also, various neighborhood degree sum-based topological indices are retrieved from the expression of these polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In addition, eight graph entropies in terms of these topologi- cal indices have been defined and calculated for Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Furthermore, some numerical values of topological indices and index-entropies of Y-junction graphs are plotted for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Since topological indices based on the degree of vertices has a significant ability to predict various physicochemical properties and biological activities of the chemical molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Therefore, the study’s findings will be a viable option for predicting various physicochemical properties and understanding the structural problems of carbon nanotube Y-junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' We mention some possible directions for future research, including multiplicative topological indices, graph index-entropies, regression models between the index-entropies and the topological indices, metric and edge metric dimension, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', to predict thermochemical data, physicochemical properties, and structural information of carbon nanotube Y-junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Data Availability No data was used to support the findings of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Conflicts of Interest There are no conflicts of interest declared by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Funding Statement The authors received no specific funding for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 19 Hβ,(J3) Hβ,(J3) Hβ,(J3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Hβ,(J3) Hβ,(J3) Hβ,(Js) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 Hβ,(J3) Hβ,(J3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Index-entropies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 [3,3] [2,2] [4,4] [5,5] 17,7] [6,6] [8,8] [9,9] [10,10] [1,m]Author’s Contribution Statement The final draft was written by Sohan Lal and Vijay Kumar Bhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Figures and Tables are prepared by Sohan Lal and Sahil Sharma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' All authors reviewed and edited the final draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Kroto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Heath, S.' metadata={'source': 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