Edge Average Wiener Index Of Graphs

Let G a connected graph. For two edges f = uv and g = xy of G, D'(f,g) = (?)(d(u,x) + d(u,y) + d(v,x) + d(v,y)) is called the average distance of / and g, and the sum∑_{{f,g} (?) E(G)}^D'(f,g) the edge average Wiener index W'_e(G) of G. It is closely related to Gutman index of graphs and Wiener index of line graphs.In this thesis, at first, we prove that the edge average Wiener index of a unicyclic graph G is an integer if and only if the length of the cycle in G is divisible by 4. We determine the maximum and minimum values of edge average Wiener index among unicyclic graphs of order n, and respectively, among connected graphs of order n and of diameter two. We also show that the star K_1,(n-1) has the minimum edge average Wiener index among all connected graphs of order n, and the path P_m+1 has the maximum edge average Wiener index among all connected graphs of size m.