On Cayley representations of central Cayley graphs over almost simple groups

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 $G$.