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 $mathcal{P}_0(G)$ of a finite group $G$, finding a formula for the number $c(mathcal{P}_0(G))$ of its components which is particularly illuminative when $Gleq S_n$ 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 $mathcal{T}_0(G)$. We show that all those graphs are quotient of $mathcal{P}_0(G)$ and demonstrate a strong link between them dealing with $G=S_n$. We find simultaneously $c(mathcal{P}_0(S_n))$ as well as the number of components of $widetilde{mathcal{P}}_0(S_n)$, $mathcal{O}_0(S_n)$ and $mathcal{T}_0(S_n)$.

Quotient graphs for power graphs / D. Bubboloni, M. A. Iranmanesh, S.M. Shaker. - In: RENDICONTI DEL SEMINARIO MATEMATICO DELL'UNIVERSITA' DI PADOVA. - ISSN 0041-8994. - STAMPA. - 138:(2017), pp. 61-89. [10.4171/RSMUP/138-3]

Quotient graphs for power graphs

BUBBOLONI, DANIELA
;
2017

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 $mathcal{P}_0(G)$ of a finite group $G$, finding a formula for the number $c(mathcal{P}_0(G))$ of its components which is particularly illuminative when $Gleq S_n$ 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 $mathcal{T}_0(G)$. We show that all those graphs are quotient of $mathcal{P}_0(G)$ and demonstrate a strong link between them dealing with $G=S_n$. We find simultaneously $c(mathcal{P}_0(S_n))$ as well as the number of components of $widetilde{mathcal{P}}_0(S_n)$, $mathcal{O}_0(S_n)$ and $mathcal{T}_0(S_n)$.
2017
138
61
89
D. Bubboloni, M. A. Iranmanesh, S.M. Shaker
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1089874
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