Unitary Cayley graphs of finite semisimple rings

Abstract

Let $R$ be a finite ring with identity $1\ne 0$. The unitary Cayley graph of $R$, denoted by $G_R$, is a graph whose vertices are elements of $R$ and $x,y$ are adjacent if and only if $x-y$ is a unit in $R$. A ring $R$ is called a $DU$-ring if for any ring $S$ such that $G_R\cong G_S$, then $R\cong S$. In this paper, we study unitary Cayley graphs of finite semisimple rings and show that finite semisimple rings of the form $M_{n_1}\left(F_{q_1}\right)\times \cdots \times M_{n_k}\left(F_{q_k}\right)$ where $M_{n_i}\left(F_{q_i}\right)\ne F_2$ for all $i\in \{1,\dots ,k\}$ are $DU$-rings. We achieve this by using the Cartesian skeleton of the graph to show that $G_{M_n\left(F_q\right)}$ is prime with respect to the direct product and deduce the result through the uniqueness of graph factorization. In addition, we show that any finite commutative reduced ring is a $DU$-ring by combining graph cancellation properties with combinatorial arguments on the eigenvalues of the unitary Cayley graph of finite semisimple rings. The two results imply that if $R$ and $S$ are rings such that $G_R\cong G_S$ and $R/J_R$ does not contain the factor $F_2$ or is a product of fields, then $R/J_R\cong S/J_S$.