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 e_G (v_i) be the eccentricity of a vertex v_i in G. We can assume that e_G (v₁) ≥ e_G (v₂) ≥ · · · ≥ e_G (v_n−1) ≥ e_G (v_n). The average eccentricity of a graph G is the mean value of eccentricities of vertices of G, 1_n∑ avec(G) = n e_G (v_i). i=1 Let γ = γ_G be the largest positive integer such that e_G (v_γG) ≥ avec(G). In this paper, we study the value of γ_G of a graph G. For any tree T of order n, we prove that 2 ≤ γ_T ≤ n − 1 and we characterize the extremal graphs. Moreover, we prove that for any graph G of order n, 2 ≤ γ_G ≤ n and we characterize the extremal graphs. Finally some Nordhaus-Gaddum type results are obtained on γ_G of general graphs G.