Quotient graphs for power graphs
Rendiconti del Seminario Matematico della Università di Padova, Tome 138 (2017), pp. 61-89.

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) of a finite group G, finding a formula for the number of its components which is particularly illuminative when GSn is a fusion controlled permutation group. We make use of the proper quotient power graph 𝒫˜0(G), the proper order graph 𝒪0(G) and the proper type graph 𝒯0(G). All those graphs are quotient of 𝒫0(G). We emphasize the strong link between them determining number and typology of the components of the above graphs for G=Sn. In particular, we prove that the power graph 𝒫(Sn) is 2-connected if and only if the type graph 𝒯(Sn) is 2-connected, if and only if the order graph 𝒪(Sn) is 2-connected, that is, if and only if either n=2 or none of n,n-1 is a prime.

Publié le :
DOI : 10.4171/RSMUP/138-3
Classification : 05, 20
Mots-clés : Quotient graph, power graph, permutation groups
Bubboloni, Daniela 1 ; Iranmanesh, Mohammad 2 ; Shaker, Seyed 2

1 Università degli Studi di Firenze, Italy
2 Yazd University, Iran
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     author = {Bubboloni, Daniela and Iranmanesh, Mohammad and Shaker, Seyed},
     title = {Quotient graphs for power graphs},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {61--89},
     publisher = {European Mathematical Society Publishing House},
     address = {Zuerich, Switzerland},
     volume = {138},
     year = {2017},
     doi = {10.4171/RSMUP/138-3},
     mrnumber = {3743245},
     zbl = {1387.05109},
     url = {https://www.numdam.org/articles/10.4171/RSMUP/138-3/}
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Bubboloni, Daniela; Iranmanesh, Mohammad; Shaker, Seyed. Quotient graphs for power graphs. Rendiconti del Seminario Matematico della Università di Padova, Tome 138 (2017), pp. 61-89. doi : 10.4171/RSMUP/138-3. https://www.numdam.org/articles/10.4171/RSMUP/138-3/

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