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 of a finite group , finding a formula for the number of its components which is particularly illuminative when is a fusion controlled permutation group. We make use of the proper quotient power graph , the proper order graph and the proper type graph . All those graphs are quotient of . We emphasize the strong link between them determining number and typology of the components of the above graphs for . In particular, we prove that the power graph is -connected if and only if the type graph is -connected, if and only if the order graph is -connected, that is, if and only if either or none of is a prime.
Cite this article
Daniela Bubboloni, Mohammad A. Iranmanesh, Seyed M. Shaker, Quotient graphs for power graphs. Rend. Sem. Mat. Univ. Padova 138 (2017), pp. 61–89DOI 10.4171/RSMUP/138-3