Average Distance and Vertex-Connectivity

The average distance p(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., mu(G) = ((n)(2))(-1) Sigma({x,y}subset of V(G)) d(G)(x, y), where V(G) denotes the vertex set of G and d(G)(x, y) is the distance between x and y. We prove that if G is a K-vertex-connected graph, kappa >= 3 an odd integer, of order n, then mu(G) <= n/2(kappa+1) + O(1). Our bound is shown to be best possible and our results, apart from answering a question of Plesnik [J Graph Theory 8 (1984), 1-24], Favaron et al. [Networks 19 (1989), 493-504], can be used to deduce a theorem that is essentially equivalent to a theorem by Egawa and Inoue [Ars Combin 51 (1999), 89-95] on the radius of a K-vertex-connected graph of given order, where kappa is odd. (C) 2009 Wiley Periodicals, Inc. J Graph Theory 62: 157-177, 2009