Given a finite group G, denote by Γ. (G) the simple undirected graph whose vertices are the (distinct) non-central conjugacy class sizes of G, and for which two vertices of Γ. (G) are adjacent if and only if they are not coprime numbers. In this note we prove that Γ. (G) is a 2-regular graph if and only if it is a complete graph with three vertices, and Γ. (G) is a 3-regular graph if and only if it is a complete graph with four vertices.
On the regularity of a graph related to conjugacy classes of groups / M. Bianchi, M. Herzog, E. Pacifici, G. Saffirio. - In: EUROPEAN JOURNAL OF COMBINATORICS. - ISSN 0195-6698. - 33:7(2012 Oct), pp. 1402-1407. [10.1016/j.ejc.2012.03.005]
On the regularity of a graph related to conjugacy classes of groups
M. Bianchi;E. Pacifici;
2012
Abstract
Given a finite group G, denote by Γ. (G) the simple undirected graph whose vertices are the (distinct) non-central conjugacy class sizes of G, and for which two vertices of Γ. (G) are adjacent if and only if they are not coprime numbers. In this note we prove that Γ. (G) is a 2-regular graph if and only if it is a complete graph with three vertices, and Γ. (G) is a 3-regular graph if and only if it is a complete graph with four vertices.
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