Vol. 9, No. 3, 2016

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Connectivity of the zero-divisor graph for finite rings

Reza Akhtar and Lucas Lee

Vol. 9 (2016), No. 3, 415–422

We study the vertex-connectivity and edge-connectivity of the zero-divisor graph ΓR associated to a finite commutative ring R. We show that the edge-connectivity of ΓR always coincides with the minimum degree, and that vertex-connectivity also equals the minimum degree when R is nonlocal. When R is local, we provide conditions for the equality of all three parameters to hold, give examples showing that the vertex-connectivity can be much smaller than minimum degree, and prove a general lower bound on the vertex-connectivity.

zero-divisor graph, connectivity, finite ring
Mathematical Subject Classification 2010
Primary: 05C25, 13A99
Received: 30 January 2015
Revised: 10 February 2015
Accepted: 4 March 2015
Published: 3 June 2016

Communicated by Scott T. Chapman
Reza Akhtar
Department of Mathematics
Miami University
Oxford, OH 45056
United States
Lucas Lee
Department of Mathematics
Miami University
Oxford, OH 45056
United States