On certain regular nicely distance-balanced graphs

TL;DR

This paper classifies regular nicely distance-balanced graphs where the parameter b3b3 equals the diameter plus one, extending the understanding of their structure beyond known cases.

Contribution

It provides a complete classification of regular nicely distance-balanced graphs with b3 = d+1, a case not previously characterized.

Findings

Regular nicely distance-balanced graphs with b3 = d+1 are classified.

Known cases include complete graphs and certain cycles for b3 = d.

The paper extends the classification to new graph families.

Abstract

A connected graph $\G$ is called {\em nicely distance--balanced}, whenever there exists a positive integer $\gamma=\gamma(\G)$, such that for any two adjacent vertices $u,v$ of $\G$ there are exactly $\gamma$ vertices of $\G$ which are closer to $u$ than to $v$, and exactly $\gamma$ vertices of $\G$ which are closer to $v$ than to $u$. Let $d$ denote the diameter of $\G$. It is known that $d \le \gamma$, and that nicely distance-balanced graphs with $\gamma = d$ are precisely complete graphs and cycles of length $2d$ or $2d+1$. In this paper we classify regular nicely distance-balanced graphs with $\gamma=d+1$.