On the average of the eccentricities of a graph

Let G = (V; E) be a simple connected graph of order n with m edges. Also let eG(vi) be the eccentricity of a vertex vi in G. We can assume that eG(v1) eG(v2) ? ... ? eG(vn-1) ? eG(vn). The average eccentricity of a graph G is the mean value of eccentricities of vertices of G, avec(G) = 1/n ?n,i=1 eG(vi). Let ? = ?G be the largest positive integer such that eG(vG ) ? avec(G). In this paper, we study the value of G of a graph G. For any tree T of order n, we prove that 2 ? ?T ? n - 1 and we characterize the extremal graphs. Moreover, we prove that for any graph G of order n,2 ? ?G ? n and we characterize the extremal graphs. Finally some Nordhaus-Gaddum type results are obtained on ?G of general graphs G.