Some Properties of the Signless Laplacian and Normalized Laplacian Tensors of General Hypergraphs

Abstract

In this paper, we obtain some properties of signless Laplacian eigenvalues of general hypergraphs. We give the upper and the lower bound of edge connectivity of general hypergraphs in terms of average degree, minimum degree, the rank and the number of vertices, or analytic connectivity α(G), respectively. We also give the upper bound of analytic connectivity α(G) of general hypergraphs in terms of the degrees of vertices. Finally, we obtain the bounds of the smallest H + -eigenvalue of the normalized Laplacian sub-tensors of general hypergraphs.

Spectral theory of weighted hypergraphs via tensors

“…For unweighted hypergraphs, K + and L + coincide with the tensors in [9]. In the case of simple graphs, the signless Kirchhoff Laplacian, signless normalized Laplacian and signless random walk Laplacian tensors coincide with the signless Kirchooff Laplacian, signless normalized Laplacian and signless random walk Laplacian matrices, respectively.…”

The characteristic polynomials of uniform double hyperstars and uniform hypertriangles

“…Just like for ordinary graphs, the spectrum of a hypergraph has a close relationship with the structure of the hypergraph. Since then, it has attracted the attention of many researchers and the spectral theory of (uniform) hypergraphs has developed rapidly, see e.g., [3,[11][12][13][14]19,[22][23][24][25].…”

Signless Laplacian Estrada index and Laplacian Estrada index of uniform hypergraphs

“…Because of the many results relating eigenvalues of graphs and structural properties, it is also natural to consider eigenvalues of uniform hypergraphs. For some general results about the spectrum of hypergraphs, see [3,12,13,25,26].…”