## 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”.

### MSC:

 05C99 Graph theory 13A99 General commutative ring theory

### Keywords:

commuting ring; zero-divisors; diameter; girth
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### References:

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