Quasi-λ-distance-balanced graphs

A graph G is said to be quasi-λ-distance-balanced if for every pair of adjacent vertices u and v, the number of vertices that are closer to u than to v is λ times bigger (or λ times smaller) than the number of vertices that are closer to v than to u, for some positive rational number λ>1. This paper introduces the concept of quasi-λ-distance-balanced graphs, and gives some interesting examples and constructions. It is proved that every quasi-λ-distance-balanced graph is triangle-free. It is also proved that the only quasi-λ-distance-balanced graphs of diameter two are complete bipartite graphs. In addition, quasi-λ-distance-balanced Cartesian and lexicographic products of graphs are characterized. Connections between symmetry properties of graphs and the metric property of being quasi-λ-distance-balanced are investigated. Several open problems are posed.