Geodesic
The Armendariz Graph of a Ring
Discussiones Mathematicae. General Algebra and Applications, Tome 38 (2018) no. 2, pp. 189-196

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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=0^na_ix^i and g(x)=∑_j=0^mb_jx^j are adjacent if and only if a_ib_j = 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 
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Abdioğlu, Cihat; Çelikel, Ece Yetkin; Das, Angsuman. The Armendariz Graph of a Ring. Discussiones Mathematicae. General Algebra and Applications, Tome 38 (2018) no. 2, pp. 189-196. http://geodesic.mathdoc.fr/item/DMGAA_2018_38_2_a1/