Some Properties Of Wiener Index And Degree Distance Of Connected Graphs

Suppose that G = (V(G),E(G)) is a connected, simple graph with vertex set V(G) and edge set E(G). The Wiener index of G is defined by W(G) = 1/2 D(v|G) and degree distance of G by D'(G) = deg_G(v)D(v|G),where deg(v) is the degree of vertex v and D(v|G) is the sum of the distances from v to all other vertices, that is D(v|G) = d(u,v). Wiener Index anddegree Distance have important effect in characterizing molecular graphs, establishing relationships between structure and properties of molecules. It is also widely used for predicting physicochemical property and biological activity.Since above indices introduced, Ivan Gutman, Ioan Tomescu and other researchers have put up more investigation with topological index([1-5]).In this thesis we discuss the properties of Wiener Index and degree Distance of connected graphs with order n. We can divide it into three chapters:In the first chapter, we investigate the unicyclic graph with maximum degree distance. Reference [3], [6] discussed the connected graph with minimum degree distance and the unicyclic graph with minimum degree distance, respectively. On the basis, we proved that the extremal graph with maximum degree distance of unicyclic graphs is obtained from a triangle C₃ by attaching a pendent path P_n-3.In the second chapter, we mainly discuss some extremal properties of Wiener index and degree distance of polycyclic graphs. Let P⁺(n) be the set of poly-cyclic graphs whose each pair of minimal cycles have no common edges and and P⁺(n,m) be the graphs with m(m ≥ 1) minimal cycles in P⁺(n). For n ≥ 7, we proved that the extremal graph with minimal degree distance in P⁺(n,m) is a follower F(n,m) which is m triangles sharing a common vertex on which n - 1 - 2m pendent edges attached . Furthermore, we proved that the extremal graph with minimal degree distance in P⁺(n) is the graph F(n, 1) .