On commuting graphs of semisimple rings

Abstract

Let R be a non-commutative ring. The commuting graph of R denoted by Γ( R), is a graph with vertex set R⧹ Z( R), and two distinct vertices a and b are adjacent if ab = ba. In this paper we investigate some properties of Γ( R), whenever R is a finite semisimple ring. For any finite field F, we obtain minimum degree, maximum degree and clique number of Γ( M n ( F)). Also it is shown that for any two finite semisimple rings R and S, if Γ( R) ≃ Γ( S), then there are commutative semisimple rings R 1 and S 1 and semisimple ring T such that R ≃ T × R 1, S ≃ T × S 1 and ∣ R 1∣ = ∣ S 1∣.