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

Abstract Given a graph G and a set X ⊆ V ( G ) , the relative Wiener index of X in G is defined as W X ( G ) = ∑ { u , v } ∈ X 2 d G ( u , v ) . 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 | V 1 | = | V 2 | we have W V 1 ( G ) = W V 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 G in which all vertices u ∈ V ( G ) have the same total distance D G ( u ) = ∑ v ∈ V ( G ) d G ( u , v ) . Some related problems are posed along the way, and the so-called Wiener game is introduced.