On the commuting graph of some non-commutative rings with unity

Abstract

‎‎Let $R$ be a non-commutative ring with unity‎. ‎The commuting graph‎ of $R$ denoted by $Gamma(R)$‎, ‎is a graph with a vertex set‎ ‎$Rsetminus Z(R)$ and two vertices $a$ and $b$ are adjacent if and only if‎ $ab=ba$‎. ‎In this paper‎, ‎we investigate non-commutative rings with unity of order $p^n$ where $p$ is prime and $n in lbrace 4,5 rbrace$‎. It is shown that‎, ‎$Gamma(R)$ is the disjoint union of complete graphs‎. Finally‎, ‎we prove that there are exactly five commuting‎ ‎graphs of non-commutative rings with unity up to twenty vertices and they are $3K_2,3K_4,7K_2‎, ‎K_2 cup 2K_6$ and $4K_2 cup K_6$‎.