Connectivity of the zerodivisor graph for finite rings Reza Akhtar and Lucas Lee Summary: Connectivity of the zero­divisor graph for finite rings Reza Akhtar and Lucas Lee 1 Introduction One way of studying the set of zero­divisors in a commutative ring R is by means of the zero­divisor graph #(R), introduced by Beck in [4] and studied further in several works since; see for example [1], [2], [3] for some general results. The vertices of #(R) are the nonzero zero­divisors of R; two vertices are adjacent if and only if the product of the ring elements they represent is zero. Philosophically, the hope is that by studying graph­theoretic properties of #(R), one may ultimately be able to draw conclusions about the structure of the zero­divisors. In this article, we study the vertex connectivity #(#(R)) and edge connectivity #(#(R)) when R is a finite ring. It is a well­known fact from elementary graph theory that (for any graph G), #(G) # #(G) # #(G), where #(G) is the minimal degree. Our results may be summarized as follows: Theorem. Let R be a finite ring and G = #(R). Then . For any R, #(G) = #(G). . If R is nonlocal, #(G) = #(G). . If R is local with maximal ideal m, let r be the index of nilpotency of m, and # = |annm| - 1. Then: Collections: Mathematics