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 with $k\geq 1$, then $\Gamma(G)$ is a complete graph with $k+1$ vertices.
Conjugacy classes of finite groups and graph regularity / M. Bianchi, R.D. Camina, M. Herzog, E. Pacifici. - In: FORUM MATHEMATICUM. - ISSN 0933-7741. - 27:6(2015 Nov), pp. 3167-3172. [10.1515/forum-2013-0098]
Conjugacy classes of finite groups and graph regularity
M. Bianchi;E. Pacifici
2015
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 with $k\geq 1$, then $\Gamma(G)$ is a complete graph with $k+1$ vertices.File in questo prodotto:
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