The zero-divisor graph of a commutative ring. (English) Zbl 0941.05062

The authors study properties of the graph \(\Gamma(R)\) of a commuting ring \(R\) (with \(1\)) defined on the set of nonzero zero-divisors with adjacency relation \((x,y)\in E\) if \(xy= 0\) noting that the class of such graphs is strongly restricted by the (commutative) ring properties of \(R\). They observe that \(\Gamma(R)\) has small diameter \((\leq 3)\) and small girth \((\leq 4)\) among other results. Other classes of algebras (e.g., BCK-algebras) with a \(0\) element permit the same definition, and produce (di)graphs of different and greater variety. An interesting problem from the graph theory point of view is to find a “best class” of algebras which permits any (di)graph \(\Gamma\) to be represented as \(\Gamma(A)\) for some \(A\) in this “best class”.


05C99 Graph theory
13A99 General commutative ring theory
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