Connectivity of the zero-divisor graph for finite rings

Akhtar, Reza; Lee, Lucas

doi:10.2140/involve.2016.9.415

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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

DOI: 10.2140/involve.2016.9.415

Abstract

We study the vertex-connectivity and edge-connectivity of the zero-divisor graph $Γ_{R}$ associated to a finite commutative ring $R$ . We show that the edge-connectivity of $Γ_{R}$ 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.

Keywords

zero-divisor graph, connectivity, finite ring

Mathematical Subject Classification 2010

Primary: 05C25, 13A99

Milestones

Received: 30 January 2015

Revised: 10 February 2015

Accepted: 4 March 2015

Published: 3 June 2016

Communicated by Scott T. Chapman

Authors

Reza Akhtar
Department of Mathematics Miami University Oxford, OH 45056 United States

Lucas Lee
Department of Mathematics Miami University Oxford, OH 45056 United States

Vol. 9, No. 3, 2016