A note on regular sets in Cayley graphs

TL;DR

This paper investigates the properties of regular sets within Cayley graphs, establishing conditions under which normal subgroups form regular sets based on parameters like regularity degree and adjacency counts.

Contribution

It provides new criteria for when normal subgroups of a finite group form regular sets in Cayley graphs, especially relating to the parity of the adjacency parameter.

Findings

If  is even, the normal subgroup is a (, au)-regular set.

If  is odd, the subgroup is a (, au)-regular set iff it is a (0,1)-regular set.

Conditions depend on subgroup normality and parameters , , and .

Abstract

A subset $R$ of the vertex set of a graph $\Gamma$ is said to be $(\kappa,\tau)$-regular if $R$ induces a $\kappa$-regular subgraph and every vertex outside $R$ is adjacent to exactly $\tau$ vertices in $R$. In particular, if $R$ is a $(\kappa,\tau)$-regular set of some Cayley graph on a finite group $G$, then $R$ is called a $(\kappa,\tau)$-regular set of $G$. Let $H$ be a non-trivial normal subgroup of $G$, and $\kappa$ and $\tau$ a pair of integers satisfying $0\leq\kappa\leq|H|-1$, $1\leq\tau\leq|H|$ and $\gcd(2,|H|-1)\mid\kappa$. It is proved that (i) if $\tau$ is even, then $H$ is a $(\kappa,\tau)$-regular set of $G$; (ii) if $\tau$ is odd, then $H$ is a $(\kappa,\tau)$-regular set of $G$ if and only if it is a $(0,1)$-regular set of $G$.