Some properties of commuting graphs of finite n-centralizer groups

Subject Areas : Algebra

Zeinab Foruzanfar ¹ , Mehdi Rezaei ^{2 *}

1 - Buein Zahra Technical University
2 - Buein Zahra Technical University

Received: 2021-12-19 Accepted : 2022-07-18 Published : 2024-01-18

Keywords: n-مرکزساز, گراف جابجایی, عنصر نامرکزی, طیف لاپلاسین,

Abstract :

Let G be a finite non-abelian group and Z(G) the center of G . The commuting graph of G , denoted by Γ_G , is a simple undirected graph whose vertex set is G-Z(G) and two distinct vertices x and y are adjacent if and only if xy=yx . We say that a group G is n - centralizer if the number of distinct centralizers of its elements is n. Also, a finite non-abelian group G is called an AC-group if C_G(x) is abelian for any x ϵG-Z(G) . The spectrum of a graph is the set of distinct eigenvalues with their multiplicities of the adjacency matrix of the graph. Similarly, the Laplacian spectrum of a graph is the set of distinct eigenvalues with their multiplicities of the Laplacian matrix of the graph. Also, the set of distinct eigenvalues with their multiplicities of the signless Laplacian matrix of the graph is called the signless Laplacian spectrum of the graph. In this paper, we show that the finite 6-centralizers and 7-centralizers groups are AC-group. Moreover, the spectrum, Laplacian spectrum and signless Laplacian spectrum of commuting graphs of these groups are computed.

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