Summary: Connectivity of the zerodivisor graph for finite rings
Reza Akhtar and Lucas Lee
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  and studied further in several
works since; see for example , ,  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:
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: