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Zero-divisor graphs of \(2 \times 2\) upper triangular matrix rings over \(\mathbb Z_n\). (English) Zbl 1308.05065

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

Citations:

Zbl 0654.13001

References:

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