Some properties of the total graph and regular graph of a commutative ring

Authors : Manal GHANEM, Khalida NAZZAL

Pages : 835-843

View : 25 | Download : 5

Publication Date : 2018-08-01

Article Type : Research Paper

Abstract :Let $R$ be a commutative ring with unity. The total graph of $R$, $Tinsert ignore into journalissuearticles values(\Gammainsert ignore into journalissuearticles values(R););$, is the simple graph with vertex set $R$ and two distinct vertices are adjacent if their sum is a zero-divisor in $R$. Let Reg $insert ignore into journalissuearticles values(\Gammainsert ignore into journalissuearticles values(R););$ and $Zinsert ignore into journalissuearticles values(\Gammainsert ignore into journalissuearticles values(R););$ be the subgraphs of $Tinsert ignore into journalissuearticles values(\Gammainsert ignore into journalissuearticles values(R););$ induced by the set of all regular elements and the set of zero-divisors in $R$, respectively. We determine when each of the graphs $Tinsert ignore into journalissuearticles values(\Gammainsert ignore into journalissuearticles values(R););$ , Reg $insert ignore into journalissuearticles values(\Gammainsert ignore into journalissuearticles values(R););$, and $Zinsert ignore into journalissuearticles values(\Gammainsert ignore into journalissuearticles values(R););$ is locally connected, and when it is locally homogeneous. When each of Reg $insert ignore into journalissuearticles values(\Gammainsert ignore into journalissuearticles values(R););$ and $Zinsert ignore into journalissuearticles values(\Gammainsert ignore into journalissuearticles values(R););$ is regular and when it is Eulerian.
Keywords : Total graph of a commutative ring, Regular graph of a commutative ring, Locally connected, Locally homogeneous, Regular graph, Eulerian graph