#### Vol. 8, No. 5, 2015

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Adjacency matrices of zero-divisor graphs of integers modulo $n$

### Matthew Young

Vol. 8 (2015), No. 5, 753–761
##### Abstract

We study adjacency matrices of zero-divisor graphs of ${ℤ}_{n}$ for various $n$. We find their determinant and rank for all $n$, develop a method for finding nonzero eigenvalues, and use it to find all eigenvalues for the case $n={p}^{3}$, where $p$ is a prime number. We also find upper and lower bounds for the largest eigenvalue for all $n$.

##### Mathematical Subject Classification 2010
Primary: 05C50, 13M99
##### Milestones
Revised: 27 August 2014
Accepted: 19 September 2014
Published: 28 September 2015

Communicated by Kenneth S. Berenhaut
##### Authors
 Matthew Young Mathematics Department Penn State University University Park 008 McAllister Building State College, PA 16802 United States