On the Construction of Singular and Cospectral Hypergraphs

In this paper, we define two operations, neighbourhood m-splitting hypergraph NSₘ(G^*) and non-neighbourhood splitting hypergraph NNS(G^*), and obtain several properties of their adjacency spectrum. We also estimate the energies of NSₘ(G^*) and NNS(G^*). Moreover, we introduce two new join operations on k-uniform hypergraphs: the neighbourhood splitting V-vertex join G₁^* G₂^* and the S-vertex join G₁^* G₂^* of hypergraphs G₁^* and G₂^*, 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.