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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
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


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


Collections: Mathematics