Monoid graphs and generalized Petersen graphs

Abstract

[en] First, a wide definition of Cayley graphs is presented. We focus on the notion of monoid graph: a graph is a monoid graph if it is isomorphic to the underlying graph of the Cayley graph $\operatorname{Cay}(M, C)$ of some monoid $M$ with some connection set $C \subseteq M$. Secondly, the family of Generalized Petersen Graphs $G(n, k)$ is presented. We study the open question whether every Generalized Petersen Graph is a monoid graph, and we focus on the smallest one for which the question remains unanswered: $G(7,2)$. Finally, we explore the feasibility of using the computer to search for a possible monoid for $G(7,2)$. We conclude that it is not viable to check all the possibilities with the proposed algorithms. Nevertheless, we are able to provide a computer-assisted proof that if $G(7,2)$ is a monoid graph then the connection set $C$ does not have any invertible element.