Study On The Structural Parameters Based On Distance Of Graphs

Only simple,undirected and connected graphs are considered in this paper.The Wiener index of a connected graphs G is defined as the sum of distances between all pairs of vertices in G,which is denoted as(?).The Wiener complexity of G is the number of different transmissions of it vertices,that is CW(G)=|{TrG(v):v?V(G)}|.Similarly,the eccentric complexity of G is defined as the number of different eccentricities of its vertices,that is Cec(G)=diam(G)-rad(G)+1.In this paper,the relations between these two complexities is study.In Section 2,the relations of CW(G)and Cec(G)are compared on Cartesian product graphs.It is shown that Cec(G)?CW(G)holds for almost all graphs.Several classes of graphs satisfy Cec(G)>CW(G)are constructed.The first and second Zagreb eccentricity indices of graph are defined as E₁(G)=?v?V(G)?_G(v)2 and E₂(G)=?uv?V(G)?_G(v)2?_G(v)respectively,where ?_G(v)denotes the eccentricity of vertex v in G.Another eccentricity-based topological index named as "eccentric connectivity index" is defined as ?c(G)=?v?V(G)degG(v)2?_G(v).In Section 3,at first,some results on the comparison between E₂(G)/m and ?c(G)/n are presented for any connected graphs G of order n with m edges.Then the Wiener index?E₁(G)and E₂(G)are compared for the graphs with diameter 2.At last,the comparison results among Wiener index,E₁(G)and E₂(G)are researched for the graphs with diameter greater than 2.