Applications On Vertex-degree, Distance-Based Topological Indices And Parameters Of Graphs

Chemical graph theory is an interdisciplinarity of graph theory and the-oretical chemistry, which applies graph theory to mathematical modelling of chemical phenomena and some chemical compounds with given physico-chemical properties. A topological index is a real number associated with chemical constitution purporting for correlation of chemical structure with various physical properties,chemical reactivity or biological activity, which be used to understand properties of chemical compounds in theoretical chem-istry [1].Wiener index W is the first distance-based topological index, introduced by American chemist Wiener for investigating boiling points of alkanes in 1947 [2]. The well known vertex-degree-based topological indices are the first and the second Zagreb indices Mi and M2,which have been intro-duced by Gutman and Trinajestic [3] and applied to study molecular, chiral-ity in quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) analysis. Up to now,hundreds of topological indices and their variants have been defined in chemical litera-tures, various applications of these topological indices have been found and many mathematical properties are also investigated, especially those vertex-degree-based and distance-based topological indices which are used for the design of chemical compounds with given physico-chemical properties or giv-en pharmacologic, toxicologic, biological activities and other properties.In this thesis, the calculation and estimation problems of some vertex-degree-based and distance-based topological indices, such as the hyper-Wiener index WW, the vertex PI index PIv, the Szeged index Sz, the weighted ver-tex PI index PIw, the Zagreb eccentricity indices M1* and M2*, the multiplica-tive Zagreb eccentricity indices â…¡1* and â…¡2* of the join, the disjunction, the symmetric difference, the composition, the Cartesian product, the hierarchi-cal product, the generalized hierarchical product, the Cluster product, the Corona product and the S-sum of graphs are investigated, and the explicit formulas for these vertex-degree-based and distance-based topological indices of some special molecular graphs are obtained by means of our main results. Moreover, we structured the Compatible System of Chinese Medicinal Ma-terials (CSCMM), then using the vertex(node)-degree-based and distance-based topological parameters of huge network(graph), such as the average vertex(node)-degree <k>, the average vertex(node)-strength <s>, the average shortest path-length <l>, the clustering coefficient C, betweenness B and so forth to analyse some quantitative and visual characteristics of CSCMM.Specifically, we have constructed our work in five chapters. A short but relatively complete introduction of this work is given in Chapter 1. First, we introduce the background and significance of this research area.Then, we give some basic definitions and notations of graphs and describe some definitions and notations of some vertex-degree-based and distance-based topological indices and graph operations. Last, we outline the main results of this thesis.In Chapter 2, the Zagreb eccentricity indices M1* and M2* of the gener-alized hierarchical product of two connected graphs G and H are computed. Moreover, we present explicit formulas for the M1* and M2* of S-sum graph, Cartesian, Cluster and Corona product graphs by means of some invariants of the factors. As applications,we present explicit formulas for the M1* and M2* indices of the C4 nanotorus Cmâ–¡Cn, the C4 nanatube Pmâ–¡Cn, the Zig-zag polyhex nanotube TUHC6[2n,2] and the hexagonal chain Ln etc.In Chapter 3, we present some bounds of the multiplicative Zagreb ec-centricity indices â…¡1*and â…¡2* of Cartesian product graphs by means of some invariants of the factors and supply some exact expressions of â…¡1* and â…¡2* indices of some composite graphs, such as the join, disjunction, symmetric difference and composition of graphs,respectively.Let G+H, G{H} and G (?) H be the join, Cluster ptoduct and Corona product of two graphs G and H,respectively. In Chapter 4, The explicit formulas for the hyper-Wiener, vertex PI, Szeged and weighted vertex PI indices of G+H, G{H} and G (?) H by means of some invariants of the factors are presented, respectively.In Chapter 5, the complex network representation of Compatible Sys-tem of Chinese Medicinal Materials (CSCMM) is structured. By means of the thought of wholism and system theory, the research methods of complex network and the technology for large network analysis and visual algorith-m. Using the average vertex(node)-degree<k>, average vertex(node)-strength (s), average shortest path-length (l), clustering coefficient C and betweenness B, density D, diameter d and radius r of a network(graph). Some quanti-tative analysis and qualitative expounds macroscopically are presented, and some properties of CSCMM, such as Small-world and approximate Scale-free and so forth are revealed. We hope to inspire and help the modern research of Chinese medicinal materials compatibility.Throughout the first four chapters of this thesis, we only consider undi-rected, simple, connected graphs. In the fifth chapter, a huge network can also be a weighted graph or a multigraph.