#### 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
##### Abstract

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