On the Construction of Singular and Cospectral Hypergraphs

TL;DR

This paper introduces new hypergraph operations and analyzes their spectral properties, leading to the construction of infinite families of singular and cospectral hypergraphs with specific spectral characteristics.

Contribution

It defines new hypergraph operations and determines their spectra, enabling the construction of singular and cospectral hypergraph families.

Findings

Defined neighbourhood m-splitting and non-neighbourhood splitting hypergraphs.

Estimated energies of the new hypergraphs.

Constructed infinite families of singular and cospectral hypergraphs.

Abstract

In this paper, we define two operations, neighbourhood m-splitting hypergraph $NS_m(\mathscr{G}^*)$ and non-neighbourhood splitting hypergraph $NNS(\mathscr{G}^*)$, and obtain several properties of their adjacency spectrum. We also estimate the energies of $NS_m(\mathscr{G}^*)$ and $NNS(\mathscr{G}^*)$. Moreover, we introduce two new join operations on $k$-uniform hypergraphs: the neighbourhood splitting V-vertex join $\mathscr{G}_1^*\veebar \mathscr{G}_2^*$ and the S-vertex join $\mathscr{G}_1^*\barwedge \mathscr{G}_2^*$ of hypergraphs $\mathscr{G}_1^*$ and $\mathscr{G}_2^*$, and determine their adjacency spectrum. As an application, we obtain infinite families of singular hypergraphs and infinite pairs of non-regular non-isomorphic cospectral hypergraphs.