Graphical regular representations of \((2, p)\)-generated groups (Q6655688)

Let \(G\) be a finite group generated by an involution and an element of odd prime order \(p\). A graph \(\Gamma\) is called a graphical regular representations (or GRR) of \(G\) if \(\mathrm{Aut}(\Gamma)\) acts regularly on the vertex set of \(\Gamma\) and is isomorphic to \(G\).\N\NIn the paper under review, the author provides some sufficient conditions for certain Cayley graphs to be a GRR. This condition allows demonstrating the existence of GRRs with prescribed valency for a broad class of groups, including some groups that are \((2,p)\)-generated. In particular this paper considers \(k\)-valent GRRs of finite nonabelian simple groups with \(k \geq 5\). The reviewer emphasizes that all nonabelian simple groups are \((2,p)\)-generated (see [\textit{C. S. H. King}, J. Algebra 478, 153--173 (2017; Zbl 1376.20018)]).