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

Abstract

The aim of this paper is to study the intersection hypergraph $\tilde{\hat{I}}{“}_{\mathcal{H}}({\mathbb{Z}}_n)$ 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 $\tilde{\hat{I}}{“}_{\mathcal{H}}({\mathbb{Z}}_n)$ and examine some of the structural properties, viz., diameter, girth and chromatic number of $\tilde{\hat{I}}{“}_{\mathcal{H}}({\mathbb{Z}}_n)$. Also, we provide characterizations for hypertrees, star structures of $\tilde{\hat{I}}{“}_{\mathcal{H}}({\mathbb{Z}}_n)$, and investigate the planarity and non-planarity of $\tilde{\hat{I}}{“}_{\mathcal{H}}({\mathbb{Z}}_n)$.