Quasi-λ-distance-balanced graphs

• Published in: Discrete Applied Mathematics Vol. 227; pp. 21 - 28
• Main Authors: Abedi, Amirabbas, Alaeiyan, Mehdi, Hujdurović, Ademir, Kutnar, Klavdija
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 20.08.2017; Elsevier BV
• Subjects: Bipartite graphs; Combinatorics; Distance-balanced graphs; Graph theory; Graphs; Quasi-[formula omitted]-distance-balanced graphs; Triangle-free graphs; Distance-balanced graphs; Triangle-free graphs; Quasi-λ-distance-balanced graphs; Bipartite graphs
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2017.04.030

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.