Zero-Divisor Graphs of $2 \times 2$ Upper Triangular Matrix Rings Over $Z_n$

Abstract

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 $n$.