Average distance and vertex-connectivity

The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = ( n 2 )−1∑ {x,y}⊂V (G) dG(x, y), where V (G) denotes the vertex set of G and dG(x, y) is the distance between x and y. We prove that if G is a κ-vertex-connected graph, κ ≥ 3 an odd integer, of order n, then μ(G) ≤ n 2(κ+1) + O(1). Our bound is shown to be best possible and our results, apart from answering a question of Plesńık [12], Favaron, Kouider and Mahéo [8], can be used to deduce a theorem that is essentially equivalent to a theorem by Egawa and Inoue [6] on the radius of a κ-vertex-connected graph of given order, where κ is odd.