JournalsrsmupVol. 138pp. 61–89

Quotient graphs for power graphs

  • Daniela Bubboloni

    Università degli Studi di Firenze, Italy
  • Mohammad A. Iranmanesh

    Yazd University, Iran
  • Seyed M. Shaker

    Yazd University, Iran
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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 P0(G)\mathcal{P}_0(G) of a finite group GG, finding a formula for the number of its components which is particularly illuminative when GSnG\leq S_n is a fusion controlled permutation group. We make use of the proper quotient power graph P~0(G)\widetilde{\mathcal{P}}_0(G), the proper order graph O0(G)\mathcal{O}_0(G) and the proper type graph T0(G)\mathcal{T}_0(G). All those graphs are quotient of P0(G)\mathcal{P}_0(G). We emphasize the strong link between them determining number and typology of the components of the above graphs for G=SnG=S_n. In particular, we prove that the power graph P(Sn)\mathcal{P}(S_n) is 22-connected if and only if the type graph T(Sn)\mathcal{T}(S_n) is 22-connected, if and only if the order graph O(Sn)\mathcal{O}(S_n) is 22-connected, that is, if and only if either n=2n = 2 or none of n,n1n, n-1 is a prime.

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Daniela Bubboloni, Mohammad A. Iranmanesh, Seyed M. Shaker, Quotient graphs for power graphs. Rend. Sem. Mat. Univ. Padova 138 (2017), pp. 61–89

DOI 10.4171/RSMUP/138-3