Quotient graphs for power graphs
D. Bubboloni, Mohammad A. Iranmanesh, S. M. Shaker

TL;DR
This paper develops a formula for counting components of the proper power graph of finite groups, especially symmetric groups, using quotient graph techniques and explores their interrelations.
Contribution
It introduces a novel application of quotient graph procedures to the proper power graph of finite groups, providing explicit formulas and insights for symmetric groups.
Findings
Derived a formula for the number of components in the proper power graph of G
Established connections between various quotient graphs of the power graph
Computed the number of components for symmetric groups S_n
Abstract
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 its quotient graphs. We apply here 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 . We show that all those graphs are quotient of and demonstrate a strong link between them dealing with . We find simultaneously as well as the number of components of , and .
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Taxonomy
TopicsFinite Group Theory Research · Limits and Structures in Graph Theory · Advanced Graph Theory Research
