The Extremal Problems Of Some Topological Indices On Several Kinds Of Cyclic Graphs

The paper aims at discussing the calculational and extremal problems of some topological indices on several classes of graphs containing cycles.Let G =(V(G),E(G))be a simple connected graph with vertex set V(G)and edge set E(G).For two arbitrary vertices u and v of a connected graph G,the distance between them,denoted by dG(u,v),is the number of edges on a shortest path joining these vertices in G.Then the Wiener index W(G),hyper-Wiener index WW(G)and Harary index H(G)of a connected graph G are respectively defined asThe Wiener,hyper-Wiener and Harary indices are three classic topological indices that have been widely investigated in chemical graph theory.They have been successfully applied in theoretical chemistry for the study of quantitative structure-property relationship(QSPR)and quantitative structure-activity relationship(QSAR).A cactus graph is a connected graph in which each block is either an edge or a cycle.A connected graph on n vertices is a bicyclic graph if it has n +1 edges.A tricyclic graph of n vertices is a connected graph with n + 2 edges.The thesis sees some conpletely new graph transformations.With the aid of these effective methods,we first identify the unique extremal graph with the largest Wiener and hyper-Wiener indices among all cacti with given vertices and cycles in the present paper.In the next place,the extremal bicyclic graphs,which have the second up to eighth largest hyper-Wiener indices,are achieved.At the end,the sharp upper bound for the Harary index,the smallest and largest Wiener and hyper-Wiener indices among tricyclic graphs are determined.Meantime,the corresponding extremal graphs are completely characterized,too.