Revelation of Zero Divisor Graph and its Multifarious Scope

Abstract

The zero-divisor graph of a non-commutative ring R can be described as the directed graph ℾ(R) whose vertices are all non-zero zero-divisors of R and

The zero-divisor graph of a non-commutative ring R can be described as the directed graph ℾ(R) whose vertices are all non-zero zero-divisors of R and in which, for any two different vertices x and y, is an edge if and only if xy=0. We look at how R's ring-theoretic and graph-theoretic aspects ℾ(R) interact. In this work, it is demonstrated that, with a finite number of exceptions, if R is a ring and S is a finite semisimple ring that is not a field and, ℾ(R)  then R  We display that if R is a ring and ℾ(R) , then R  By putting off all instructions from the edges in Redmond's definition of the easy undirected design ℾ(R). We categorise any ring R whose (R) as both a whole graph, a bipartite graph, and a tree.

in which, for any two different vertices x and y, is an edge if and only if xy=0. We look at how R's ring-theoretic and graph-theoretic aspects ℾ(R) interact. In this work, it is demonstrated that, with a finite number of exceptions, if R is a ring and S is a finite semisimple ring that is not a field and, ℾ(R)  then R  We display that if R is a ring and ℾ(R) , then R  By putting off all instructions from the edges in Redmond's definition of the easy undirected design ℾ(R). We categorise any ring R whose (R) as both a whole graph, a bipartite graph, and a tree.