The Armendariz Graph of a Ring

Abstract

In this paper we initiate the study of Armendariz graph of a commutative ring R and investigate the basic properties of this graph such as diameter, girth, domination number, etc. The Armendariz graph of a ring R, denoted by A(R), is an undirected graph with nonzero zero-divisors of R[x] (i.e., Z(R[x])*) as the vertex set, and two distinct vertices f(x)=∑i=0naixi$f(x) = \sum\nolimits_{i = 0}^n {{a_i}{x^i}}$ and g(x)=∑j=0mbjxj$g(x) = \sum\nolimits_{j = 0}^m {{b_j}{x^j}}$ are adjacent if and only if aibj = 0, for all i, j. It is shown that A(R), a subgraph of Γ(R[x]), the zero divisor graph of the polynomial ring R[x], have many graph properties in common with Γ(R[x]).

Keywords

• armendariz property
• diameter
• girth
• zero-divisor graph
• 05c12
• 05c25