Spectrum and genus of commuting graphs of some classes of finite rings

Abstract

The commuting graph of a non-commutative ring R with center Z(R) is a simple undirected graph whose vertex set is R \ Z(R) and two vertices x, y are adjacent if and only if xy = yx. In this paper, we compute the spectrum and genus of commuting graphs of some classes of finite rings.

On r-Noncommuting Graph of Finite Rings

“…Recall that the noncommuting graph of a finite noncommutative ring R is a simple undirected graph whose vertex set is R \ Z(R) and two vertices x and y are adjacent if and only if xy = yx. The complement of noncommuting graph, called commuting graph, of a finite noncommutative ring is considered in [2][3][4][5]. The motivation for studying commuting/noncommuting graphs of finite rings comes from the study of commuting/noncommuting graphs of finite groups.…”

Various spectra and energies of commuting graphs of finite rings

“…The commuting graph of R, denoted by Γ R , is a simple undirected graph whose vertex set is R \ Z(R) and two vertices x, y are adjacent if and only if xy = yx. In recent years, many mathematicians have considered commuting graph of different rings and studied various graph theoretic aspects (see [1,3,12,13,17,19,20,23]). Some generalizations of Γ R are also considered in [2,9].…”

“…, S n of non-central elements of R. Then E(Γ R ) = 2(|R| − |Z(R)| − n).Proof. By Theorem 2.1 of[12] we haveSpec(Γ R ) = {(−1) n i=1 |S i |−n(|Z(R)|+1) , (|S 1 | − |Z(R)| − 1) 1 , . .…”

On Generalized Non-commuting Graph of a Finite Ring

“…Note that Z(R) = ∩ r∈R C R (r). Many mathematicians have studied algebraic structures by means of graph theoretical properties in the last decades (see [1,2,3,4,9,14] etc.). The notion of non-commuting graph of a finite ring was introduced by Erfanian, Khashyarmanesh and Nafar [10].…”