The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs

H-eigenvalues of Laplacian and Signless Laplacian Tensors

We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H-eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, and H-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor, and the adjacency tensor has at most one H-eigenva...

The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph

In this paper, we show that the eigenvectors of the zero Laplacian and signless Lapacian eigenvalues of a k-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector of the zero Laplacian or signless Lapacian eigenvalue have the same modulus. Moreover, under a canonical regularization, the phases of the components of t...

Regular Uniform Hypergraphs, s-Cycles, s-Paths and Their largest Laplacian H-Eigenvalues∗

In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected k-uniform hypergraph G, where k ≥ 3, reaches its upper bound 2∆(G), where ∆(G) is the largest degree of G, if and only if G is regular. Thus the largest Laplacian H-eigenvalue of G, reaches the same upper bound, if and only if G is regular and oddbipartite. We show that an s-cycle G, as a k-uniform hypergraph...

Bibliography on the Signless Laplacian Eigenvalues

Signless Laplacian eigenvalues and circumference of graphs

In this paper, we investigate the relation between the Q -spectrum and the structure of G in terms of the circumference of G. Exploiting this relation, we give a novel necessary condition for a graph to be Hamiltonian by means of its Q -spectrum. We also determine the graphs with exactly one or two Q -eigenvalues greater than or equal to 2 and obtain all minimal forbidden subgraphs and maximal ...