Quotient graphs for power graphs

Bubboloni, Daniela; Iranmanesh, Mohammad; Shaker, Seyed

Rendiconti del Seminario Matematico della Università di Padova, Tome 138 (2017), pp. 61-89.

In a previous paper of the first author a procedure was developed for counting the components of a graph through the knowledge of the components of one of its quotient graphs. Here we apply that procedure to the proper power graph ${𝒫}_0\left(G\right)$ of a finite group $G$, finding a formula for the number of its components which is particularly illuminative when $G\le S_n$ is a fusion controlled permutation group. We make use of the proper quotient power graph ${\widetilde{𝒫}}_0\left(G\right)$, the proper order graph ${𝒪}_0\left(G\right)$ and the proper type graph ${𝒯}_0\left(G\right)$. All those graphs are quotient of ${𝒫}_0\left(G\right)$. We emphasize the strong link between them determining number and typology of the components of the above graphs for $G=S_n$. In particular, we prove that the power graph $𝒫\left(S_n\right)$ is $2$-connected if and only if the type graph $𝒯\left(S_n\right)$ is $2$-connected, if and only if the order graph $𝒪\left(S_n\right)$ is $2$-connected, that is, if and only if either $n=2$ or none of $n,n-1$ is a prime.

Classification : 05, 20
Mots clés : Quotient graph, power graph, permutation groups