On $\ell$-distance-balancedness of cubic Cayley graphs of dihedral groups

TL;DR

This paper investigates the $ ext{ell}$-distance-balanced property of cubic Cayley graphs of dihedral groups, proving that certain generating sets produce highly distance-balanced graphs, thus advancing understanding of their structural symmetry.

Contribution

It demonstrates that cubic Cayley graphs generated by specific sets are highly distance-balanced, partially addressing a problem posed by Miklavič and Šparl.

Findings

Certain cubic Cayley graphs are highly distance-balanced.

The results apply to graphs generated by specific sets involving dihedral groups.

Partial solution to an open problem in graph symmetry.

Abstract

A connected graph $\Gamma$ of diameter ${\rm diam}(\Gamma) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}(\Gamma)|=|W_{yx}(\Gamma)|$ for every $x,y\in V(\Gamma)$ with $d_{\Gamma}(x,y)=\ell$, where $W_{xy}(\Gamma)$ is the set of vertices of $\Gamma$ that are closer to $x$ than to $y$. $\Gamma$ is said to be highly distance-balanced if it is $\ell$-distance-balanced for every $\ell\in [{\rm diam}(\Gamma)]$. It is proved that every cubic Cayley graph whose generating set is one of $\{a,a^{n-1},ba^r\}$ and $\{a^k,a^{n-k},ba^t\}$ is highly distance-balanced. This partially solves a problem posed by Miklavi\v{c} and \v{S}parl.