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

• A new characterization of distance balanced graph of even order under the name “Equal Opportunity Graphs”.
• Construction of a new infinite family of distance balanced partial cubes.
• Introduction of a new game played on the vertices of a graphs named as “Wiener game”.

Given a graph GG and a set X⊆V(G)X⊆V(G), the relative Wiener index of XX in GG is defined as WX(G)=∑{u,v}∈X2dG(u,v). The graphs GG (of even order) in which for every partition V(G)=V1+V2V(G)=V1+V2 of the vertex set V(G)V(G) such that |V1|=|V2||V1|=|V2| we have WV1(G)=WV2(G)WV1(G)=WV2(G) are called equal opportunity graphs. In this note we prove that a graph GG 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 GG in which all vertices u∈V(G)u∈V(G) have the same total distance DG(u)=∑v∈V(G)dG(u,v)DG(u)=∑v∈V(G)dG(u,v). Some related problems are posed along the way, and the so-called Wiener game is introduced.