Connectivity of the zero-divisor graph for finite rings Reza Akhtar and Lucas Lee Summary: Connectivity of the zero-divisor graph for finite rings Reza Akhtar and Lucas Lee Abstract Let R be a finite ring; the authors study the vertex- and edge-connectivity of the zero-divisor graph (R). It is shown that the edge-connectivity of (R) always coincides with the minimum degree, and lower and upper bounds are given for the vertex connectivity. Conditions are given for the coincidence of these bounds, and several examples are given showing that the bounds are not always achieved. 1 Introduction One way of studying the set of zero-divisors in a commutative ring R is by means of the zero-divisor 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 zero-divisors 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 graph-theoretic properties of (R), one may ultimately be able to draw conclusions about the structure of the zero-divisors. In this article, we study the vertex connectivity ((R)) and edge connectivity ((R)) when R is a finite ring. It is a well-known fact from elementary graph theory that (for any graph G), (G) (G) (G), where (G) is the minimal degree. Our Collections: Mathematics