Structure in Zero-Divisor Graphs of Commutative Rings

In this research, we associate a graph in a natural way with the zero-divisors of a commutative ring. We endeavor to characterize various attributes of the graph, including connectivity, diameter, and symmetry. In exploring symmetry in the graph, we examine the automorphism group of the graph, and provide a complete characterization for the rings Z_N. Secondly, we seek ring-theoretic properties which may be described in terms of the associated zero-divisor graph. These include, among other results, a strong relationship between finite local rings and graphs admitting a vertex connected to every other vertex.