On Intersection and Co-maximal Hypergraph of ${\mathbb{Z}}_n$

The aim of this paper is to study the intersection hypergraph ${\tilde{\mathrm{\Gamma }}}_{\mathcal{H}}({\mathbb{Z}}_n)$ and co-maximal hypergraph $C{o}_{\mathcal{H}}({\mathbb{Z}}_n)$ on the subgroups of ${\mathbb{Z}}_n$. We prove that the intersection and co-maximal hypergraph of a finite abelian group are isomorphic. Hence, we focus on ${\tilde{\mathrm{\Gamma }}}_{\mathcal{H}}({\mathbb{Z}}_n)$ and examine some of the structural properties, viz., diameter, girth and chromatic number of ${\tilde{\mathrm{\Gamma }}}_{\mathcal{H}}({\mathbb{Z}}_n)$. Also, we provide characterizations for hypertrees, star structures of ${\tilde{\mathrm{\Gamma }}}_{\mathcal{H}}({\mathbb{Z}}_n)$, and investigate the planarity and non-planarity of ${\tilde{\mathrm{\Gamma }}}_{\mathcal{H}}({\mathbb{Z}}_n)$.