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 𝒫 0 (G) of a finite group G, finding a formula for the number of its components which is particularly illuminative when GS n is a fusion controlled permutation group. We make use of the proper quotient power graph 𝒫 ˜ 0 (G), the proper order graph 𝒪 0 (G) and the proper type graph 𝒯 0 (G). All those graphs are quotient of 𝒫 0 (G). We emphasize the strong link between them determining number and typology of the components of the above graphs for G=S n . In particular, we prove that the power graph 𝒫(S n ) is 2-connected if and only if the type graph 𝒯(S n ) is 2-connected, if and only if the order graph 𝒪(S n ) 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 = {http://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. http://www.numdam.org/articles/10.4171/RSMUP/138-3/

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