The power serieswise Armendariz graph of a commutative ring

Abstract

The rings considered in this article are commutative with non-zero identity which are not integral domains. Let R be a ring. Let Z(R) denote the set of all zero-divisors of R and we denote Z(R)\{0} by Z(R)∗. In this article, we introduce and investigate the power serieswise Armendariz graph of R denoted by PA(R). It is the undirected graph whose vertex set is Z(R[[X]])∗ and distinct vertices f(X)=∑i=0∞ aiXi and g(X)=∑j=0∞ bjXj are adjacent in PA(R) if and only if ai bj = 0 for all i and j. The aim of this article is to study the interplay between the ring-theoretic properties of R and the graph-theoretic properties of PA(R). We discuss some results on diameter, clique, and girth of PA(R).