Given a finite group G, denote by Gamma(G) the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of G, and set two vertices of Gamma(G) to be adjacent if and only if they are not coprime numbers. In this note we prove that, if Gamma(G) is a k-regular graph and either G is an F-group or k <= 5, then Gamma(G) is a complete graph with k + 1 vertices.

On the regularity of a graph related to conjugacy classes of groups: small valencies / M. Bianchi, M. Herzog, E. Pacifici (CONTEMPORARY MATHEMATICS). - In: Group theory, combinatorics, and computing / [a cura di] R. F. Morse, D. Nikolova-Popova, S. Witherspoon. - Providence : American Mathematical Society, 2014. - ISBN 9780821894354. - pp. 1-7 (( convegno International Conference in Honor of Daniela Nikolova-Popova's 60th Birthday on Group Theory, Combinatorics and Computing tenutosi a Boca Raton nel 2012 [10.1090/conm/611/12203].

On the regularity of a graph related to conjugacy classes of groups: small valencies

M. Bianchi
;
E. Pacifici
Ultimo
2014

Abstract

Given a finite group G, denote by Gamma(G) the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of G, and set two vertices of Gamma(G) to be adjacent if and only if they are not coprime numbers. In this note we prove that, if Gamma(G) is a k-regular graph and either G is an F-group or k <= 5, then Gamma(G) is a complete graph with k + 1 vertices.
Finite groups; conjugacy class sizes; classes graphs; F-groups
Settore MAT/02 - Algebra
Florida Atlantic University
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/251020
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