On Cayley representations of central Cayley graphs over almost simple groups

TL;DR

This paper investigates the uniqueness and enumeration of Cayley representations of central Cayley graphs over simple and almost simple groups, providing theoretical bounds and an efficient algorithm for their classification.

Contribution

It proves bounds on the number of Cayley representations over simple groups and introduces a polynomial-time algorithm for classifying all representations over almost simple groups.

Findings

Central Cayley graphs over simple groups have at most two nonequivalent representations.

An algorithm is provided to find all representations over almost simple groups with bounded socle index.

The algorithm runs in polynomial time relative to the size of the group.

Abstract

A Cayley graph over a group $G$ is said to be central if its connection set is a normal subset of $G$. We prove that every central Cayley graph over a simple group $G$ has at most two pairwise nonequivalent Cayley representations over $G$ associated with the subgroups of $Sym(G)$ induced by left and right multiplications of $G$. We also provide an algorithm which, given a central Cayley graph $\Gamma$ over an almost simple group $G$ whose socle is of a bounded index, finds the full set of pairwise nonequivalent Cayley representations of $\Gamma$ over $G$ in time polynomial in size of $G$.