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November 2014 Zero-Divisor Graphs of $2 \times 2$ Upper Triangular Matrix Rings Over $Z_n$
Todd Fenstermacher, Ethan Gegner
Missouri J. Math. Sci. 26(2): 151-167 (November 2014). DOI: 10.35834/mjms/1418931956


Istvan Beck introduced the zero-divisor graph in 1988. We explore the directed and undirected zero-divisor graphs of the rings of $2 \times 2$ upper triangular matrices mod $n$, denoted by $\Gamma(T_2(n))$ and $\tilde{\Gamma}(T_2(n))$, respectively. For prime $p$, we completely characterize the graph $\Gamma(T_2(p))$ by partitioning $T_2(p)$, and prove several key properties of the graphs using this approach. We establish additional properties of $\Gamma(T_2(n))$ for arbitrary $n$. We prove that $\Gamma(T_2(n))$ is Hamiltonian if and only if $n$ is prime, and we give explicit formulas for the edge connectivity and clique number of $\Gamma(T_2(n))$ in terms of the prime factorization of $n$.


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Todd Fenstermacher. Ethan Gegner. "Zero-Divisor Graphs of $2 \times 2$ Upper Triangular Matrix Rings Over $Z_n$." Missouri J. Math. Sci. 26 (2) 151 - 167, November 2014.


Published: November 2014
First available in Project Euclid: 18 December 2014

zbMATH: 1308.05065
MathSciNet: MR3293812
Digital Object Identifier: 10.35834/mjms/1418931956

Primary: 05C40 , 05C45 , 05C69
Secondary: 16S50

Keywords: Clique number , edge-connectivity , Hamiltonian , non-commutative ring , upper-triangular matrices , zero-divisor graph

Rights: Copyright © 2014 Central Missouri State University, Department of Mathematics and Computer Science


Vol.26 • No. 2 • November 2014
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