## Central sets and radii of the zero-divisor graphs of commutative rings.(English)Zbl 1105.13007

Given a commutative ring $$R$$ with identity, the zero-divisor graph of $$R$$ is the graph $$\Gamma(R)$$ where the vertices are the nonzero zero-divisors, and where there is an undirected edge between two vertices $$x$$ and $$y$$ if $$xy=0$$. This graph and its properties have been examined by a number of different authors.
The diameter of a connected graph $$G$$ is the maximum distance between any two distinct vertices. The radius of $$G$$ is the minimum of $$\{e(x): x$$ a vertex in $$G \}$$, where $$e(x)$$ denotes the maximum distance from $$x$$ to any other vertex of $$G$$. The center of $$G$$ is the set of vertices $$x$$, such that $$e(x)$$ equals the radius. It is well known that for any graph $$G$$ the diameter is bounded by twice the radius.
It has been shown that for any commutative ring $$R$$, if $$\Gamma(R)$$ is connected, then the graph has diameter at most 3 and all numbers between 0 and 3 are possible.
The current article gives further descriptions of $$\Gamma(R)$$ in terms of the above parameters under certain assumption on $$R$$. If $$R$$ is noetherian and not an integral domain, then the radius of $$\Gamma(R)$$ is at most 2. As a corollary, if $$R$$ is noetherian and not an integral domain, then there exist a nonzero $$x\in R$$ such that either $$xy=0$$ or ann$$(x)\cap$$ ann$$(y)\neq\{0\}$$ for all zero-divisors $$y\in R$$.
The center of a ring $$R$$ can be described (up to several options) in certain cases, such as when $$R$$ is an Artinian ring. Furthermore, when $$R$$ is Artinian the radius can be determined from the diameter. A dominating set of $$\Gamma(R)$$ is constructed using elements of the center when $$R$$ is Artinian.

### MSC:

 13A99 General commutative ring theory 05C99 Graph theory 13M99 Finite commutative rings
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### References:

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