Quotient graphs for power graphs
Daniela Bubboloni, Mohammad Ali Iranmanesh, Seyed Mostafa Shaker · Rendiconti del Seminario Matematico della Università di Padova · 2017
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 \mathcal{P}_0(G) of a finite group G , finding a formula for the number of its components which is particularly illuminative when G\leq S_n is a fusion controlled permutation group. We make use of the proper quotient power graph \widetilde{\mathcal{P}}_0(G) , the proper order graph \mathcal{O}_0(G) and the proper type graph \mathcal{T}_0(G) . All those graphs are quotient of \mathcal{P}_0(G) . 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 \mathcal{P}(S_n) is 2 -connected if and only if the type graph \mathcal{T}(S_n) is 2 -connected, if and only if the order graph \mathcal{O}(S_n) is 2 -connected, that is, if and only if either n = 2 or none of n, n-1 is a prime.