On the second-largest modulus among the eigenvalues of a power hypergraph

It is well known that the algebraic multiplicity of an eigenvalue of a graph (or real symmetric matrix) is equal to the dimension of its corresponding linear eigen-subspace, also known as the geometric multiplicity. However, for hypergraphs, the relationship between these two multiplicities remains an open problem. For a graph G = ( V , E ) and k ≥ 3 , the k -power hypergraph G ( k ) is a k -uniform hypergraph obtained by adding k − 2 new vertices to each edge of G , who always has non-real eigenvalues. In this paper, we determine the second-largest modulus Λ among the eigenvalues of G ( k ) , which is indeed an eigenvalue of G ( k ) . The projective eigenvariety V Λ associated with Λ is the set of the eigenvectors of G ( k ) corresponding to Λ considered in the complex projective space. We show that the dimension of V Λ is zero, i.e., there are finitely many eigenvectors corresponding to Λ up to a scalar. We give both the algebraic multiplicity of Λ and the total multiplicity of the eigenvector in V Λ in terms of the number of the weakest edges of G . Our results show that these two multiplicities are equal.