On the zero-divisor hypergraph of a reduced ring

Abstract

The concept of zero-divisor graphs of rings is widely used for establishing relationships between the properties of graphs and the properties of the underlying ring. The zero-divisor graph of a ring is generalized to the k-zero-divisor hypergraph of a ring R for \(k\in \mathbb{N}\), which is denoted by \(\mathcal{H}_{k}(R)\). This paper is an endeavor to discuss some properties of zero-divisor hypergraphs. We determine the diameter and girth of \(\mathcal{H}_{k}(R)\) whenever R is reduced. Also, we characterize all commutative rings R for which \(\mathcal{H}_{k}(R)\) is in some known class of graphs. Further, we obtain certain necessary conditions for \(\mathcal{H}_{k}(R)\) to be a Hamilton Berge cycle and a flag-traversing tour. Moreover, we answer a case of the question raised by Eslahchi et al. [15].