On the Construction of Singular and Cospectral Hypergraphs

Abstract

In this paper, we define two operations, neighbourhood m-splitting hypergraph $N{S}_m({\mathcal{G}}^{\ast })$ and non-neighbourhood splitting hypergraph $NNS({\mathcal{G}}^{\ast })$, and obtain several properties of their adjacency spectrum. We also estimate the energies of $N{S}_m({\mathcal{G}}^{\ast })$ and $NNS({\mathcal{G}}^{\ast })$. Moreover, we introduce two new join operations on $k$-uniform hypergraphs: the neighbourhood splitting V-vertex join ${\mathcal{G}}_1^{\ast }⊻{\mathcal{G}}_2^{\ast }$ and the S-vertex join ${\mathcal{G}}_1^{\ast }⊼{\mathcal{G}}_2^{\ast }$ of hypergraphs ${\mathcal{G}}_1^{\ast }$ and ${\mathcal{G}}_2^{\ast }$, 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.