Minimum Distance-Unbalancedness of Graphs With Diameter 2 and Given Number of Edges

Abstract

For a graph G, and for two distinct vertices u and v of G, let nG(u, v) be the number of vertices of G that are closer in G to u than to v. The distance-unbalancedness of G is the sum of |nG(u, v)− nG(v, u)| over all unordered pairs of distinct vertices u and v of G. We determine the minimum distance-unbalancedness of 2-self-centered graphs with given number of edges. We also determine the minimum distance-unbalancedness of graphs with at least one universal vertex and given number of edges.