Fenstermacher, Todd; Gegner, Ethan Zero-divisor graphs of \(2 \times 2\) upper triangular matrix rings over \(\mathbb Z_n\). (English) Zbl 1308.05065 Missouri J. Math. Sci. 26, No. 2, 151-167 (2014). Summary: I. Beck [J. Algebra 116, No. 1, 208–226 (1988; Zbl 0654.13001)] introduced the zero-divisor graph. We explore the directed and undirected zero-divisor graphs of the rings of \(2 \times 2\) upper triangular matrices \(\bmod n\), 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 \(n\). MSC: 05C40 Connectivity 05C45 Eulerian and Hamiltonian graphs 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 16S50 Endomorphism rings; matrix rings 11A51 Factorization; primality Keywords:non-commutative ring; zero-divisor graph; clique number; edge-connectivity; Hamiltonian; upper-triangular matrices Citations:Zbl 0654.13001 × Cite Format Result Cite Review PDF Full Text: Euclid References: [1] D. D. Anderson and M. Naseer, Beck’s coloring of a commutative ring , Journal of Algebra, 159.2 (1993), 500-514. · Zbl 0798.05067 · doi:10.1006/jabr.1993.1171 [2] D. F. Anderson, M. C. Axtell, and J. A. Stickles, Zero-divisor graphs in commutative rings , Commutative Algebra, Springer, New York, 2011. [3] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring , Journal of Algebra, 217.2 (1999), 434-447. · Zbl 0941.05062 · doi:10.1006/jabr.1998.7840 [4] M. Artin, Algebra , Prentice Hall, New Jersey, 1991. [5] I. Beck, Coloring of commutative rings , Journal of Algebra, 116.1 (1988), 208-226. · Zbl 0654.13001 · doi:10.1016/0021-8693(88)90202-5 [6] M. Behboodi and R. Beyranvand, Strong zero-divisor graphs of non-commutative rings , International Journal of Algebra, 2.1 (2008), 25-44. · Zbl 1146.16014 [7] B. Bollabás, Graph Theory, An Introductory Course , Springer-Verlag, New York, 1979. [8] R. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction , Addison-Wesley, Longman, Mass., 1998. [9] I. N. Herstein, Topics in Algebra , Blaisdell Pub., New York, 1964. · Zbl 0122.01301 [10] T. Y. Lam, A First Course in Noncommutative Rings , Springer-Verlag, New York, 1991. · Zbl 0728.16001 [11] I. Niven, H. Montgomery, and H. Zuckerman, An Introduction to the Theory of Numbers , Wiley, New York, 1991. · Zbl 0723.11001 [12] S. Redmond, The zero-divisor graph of a non-commutative ring , Trends in Commutative Rings Research, Nova Science, New York, 2004. · Zbl 1195.16038 [13] S. Redmond, On zero-divisor graphs of small finite commutative rings , Discrete Mathematics, 307.21 (2007), 1155-1166. · Zbl 1107.13006 · doi:10.1016/j.disc.2006.07.025 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.