ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II

Authors : Mohammad ABDULLA, Ayman BADAWI

Pages : 61-74

Doi:10.24330/ieja.768135

View : 15 | Download : 9

Publication Date : 2020-07-14

Article Type : Research Paper

Abstract :In 2015, the second-named author introduced the dot product graph associated to a commutative ring $A$. Let $A$ be a commutative ring with nonzero identity, $1 \leq n < \infty$ be an integer, and $R = A \times A \times \cdots \times A$ insert ignore into journalissuearticles values($n$ times);. We recall that the total dot product graph of $R$ is the insert ignore into journalissuearticles values(undirected); graph $TDinsert ignore into journalissuearticles values(R);$ with vertices $R^* = R\setminus \{insert ignore into journalissuearticles values(0, 0, \dots, 0);\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x\cdot y = 0 \in A$ insert ignore into journalissuearticles values(where $x\cdot y$ denotes the normal dot product of $x$ and $y$);. Let $Zinsert ignore into journalissuearticles values(R);$ denote the set of all zero-divisors of $R$. Then the zero-divisor dot product graph of $R$ is the induced subgraph $ZDinsert ignore into journalissuearticles values(R);$ of $TDinsert ignore into journalissuearticles values(R);$ with vertices $Zinsert ignore into journalissuearticles values(R);^* = Zinsert ignore into journalissuearticles values(R); \setminus \{insert ignore into journalissuearticles values(0, 0, \dots, 0);\}$. Let $Uinsert ignore into journalissuearticles values(R);$ denote the set of all units of $R$. Then the unit dot product graph of $R$ is the induced subgraph $UDinsert ignore into journalissuearticles values(R);$ of $TDinsert ignore into journalissuearticles values(R);$ with vertices $Uinsert ignore into journalissuearticles values(R);$. In this paper, we study the structure of $TDinsert ignore into journalissuearticles values(R);$, $UDinsert ignore into journalissuearticles values(R);$, and $ZDinsert ignore into journalissuearticles values(R);$ when $A = Z_n$ or $A = GFinsert ignore into journalissuearticles values(p^n);$, the finite field with $p^n$ elements, where $n \geq 2$ and $p$ is a prime positive integer.
Keywords : Dot product graph, annihilator graph, total graph, zero divisor graph