Equal opportunity networks, distance-balanced graphs, and Wiener game

Given a graph G and a set X subset of V(G), the relative Wiener index of X in G is defined as W-x (G) = Sigma{u, upsilon}(is an element of) (x/2) d(G)(u, upsilon). The graphs G (of even order) in which for every partition V (G) = V-1 + V-2 of the vertex set V (G) such that vertical bar V-1 vertical bar = vertical bar V-2 vertical bar we have Wv(1) (G) = Wv(2) (G) are called equal opportunity graphs. In this note we prove that a graph G of even order is an equal opportunity graph if and only if it is a distance-balanced graph. The latter graphs are known by several characteristic properties, for instance, they are precisely the graphs Gin which all vertices u epsilon V (G) have the same total distance D-G(u) = Sigma(upsilon is an element of V(G)) d(G)(u, upsilon). Some related problems are posed along the way, and the so-called Wiener game is introduced. (C) 2014 Elsevier B.V. All rights reserved.