On a conjecture about the commuting graphs of finite matrix rings

Abstract

Let R be a noncommutative ring with unity. The commuting graph of R, denoted by $\mathrm{\Gamma }(R)$, is a graph whose vertices are noncentral elements of R and two distinct vertices x and y are adjacent if $xy=yx$. Let F be a finite field and $n\ge 2$. It is conjectured by Akbari, Ghandehari, Hadian and Mohammadian in 2004 that if $\mathrm{\Gamma }(R)\cong \mathrm{\Gamma }(M_n(F))$, then $R\cong M_n(F)$. In this paper, we prove the conjecture whenever n is of the form $2^k{3}^l$ with $k\ne 0$.