Central sets and radii of the zero-divisor graphs of commutative rings

For a commutative ring R with identity, the zero-divisor graph, Gamma(R), is the graph with vertices the nonzero zero-divisors of R and edges between distinct vertices x and y whenever xy = 0. This article gives a proof that the radius of Gamma(R) is 0, 1, or 2 if R is Noetherian. The center union {0} is shown to be a union of annihilator ideals if R is Artinian. The diameter of Gamma(R) can be determined once the center is identified. If R is finite, then the median is shown to be a subset of the center. A dominating set of Gamma(R) is constructed using elements of the center when R is Artinian. It is shown that for a finite ring R not congruent to Z(2) x F for some finite field F , the domination number of Gamma(R) is equal to the number of distinct maximal ideals of R . Other results on the structure of Gamma(R) are also presented.