On the Laplacian spectrum of k-uniform hypergraphs

Abstract

In this article, we consider the generalization of the Laplacian matrix for hypergraphs to obtain several results related to spectral properties of hypergraphs. This generalization of Laplacian matrix was defined in 2021 by Anirban Banerjee. We first supply a necessary and sufficient condition on the Laplacian spectral radius of hypergraphs such that the complement of that hypergraph is connected. We give bounds for the Laplacian spectral radius of k-uniform hypergraphs in terms of some invariants of hypergraph, such as the maximum degree, minimum degree, first Zagreb index, and the chromatic number. As an application of these results, some upper bounds of the Nordhaus-Gaddum type are obtained for the sum of Laplacian spectral radius of a k-uniform hypegraphs and its complement, and product of Laplacian spectral radius of a k-uniform hypegraphs and its complement. It is known that the second largest Laplacian eigenvalue for graphs is greater or equal to the second highest degree of graphs. We show by an example that this result is not true for 3-uniform hypergraphs in general. Finally, we supply a class of 3-uniform hypergraphs for which the second largest Laplacian eigenvalue is greater or equal to the second highest degree.