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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:

  

Source: Akhtar, Reza - Department of Mathematics and Statistics, Miami University (Ohio)

 

Collections: Mathematics