Quasi-({\lambda}; n)-distance-balanced graphs

TL;DR

This paper introduces and studies quasi-({\lambda}; n)-distance-balanced graphs, providing properties, construction formulas for any diameter, and discussing related problems and conjectures in the field.

Contribution

It defines a new class of graphs, quasi-({\lambda}; n)-distance-balanced, and offers methods to construct such graphs for any diameter, extending previous work on quasi-\lambda-DB graphs.

Findings

Established properties of quasi-({\lambda}; n)-distance-balanced graphs.

Presented a formula for constructing these graphs for arbitrary diameters.

Discussed open problems and conjectures related to the class.

Abstract

For every pair of vertices u and v with d(u; v) = n, Wun G v denotes the set of all vertices of G that are closer to u than to v. In this paper, we introduce quasi-({\lambda}; n)-distance-balanced graphs and then study some properties of these graphs and present a formula to construct such graphs for arbitrarily diameter d. For n = 1, this class of graphs contains the quasi-{\lambda}-DB graphs recently introduced by Abedi et al. [Quasi-{\lambda}-distance-balanced graphs, Discrete Appl. Math. 227 (2017) 21{28]. Moreover, we will take a look at the problems arisen by Abedi et al. Some problems and conjecture are involved.