Graphical regular representations of $(2,p)$-generated groups

TL;DR

This paper establishes a sufficient condition for certain Cayley graphs of $(2,p)$-generated groups to be graphical regular representations, allowing the construction of GRRs with specified valency for a broad class of groups.

Contribution

It introduces a new criterion for Cayley graphs to be GRRs and demonstrates the existence of $k$-valent GRRs for finite nonabelian simple groups with $k \,\geq\, 5$.

Findings

Provided a sufficient condition for Cayley graphs to be GRRs.

Proved the existence of $k$-valent GRRs for certain finite simple groups.

Extended the class of groups known to admit graphical regular representations.

Abstract

For groups $G$ that can be generated by an involution and an element of odd prime order, this paper gives a sufficient condition for a certain Cayley graph of $G$ to be a graphical regular representation (GRR), that is, for the Cayley graph to have full automorphism group isomorphic to $G$. This condition enables one to show the existence of GRRs of prescribed valency for a large class of groups, and in this paper, $k$-valent GRRs of finite nonabelian simple groups with $k\geq5$ are considered.