Eigenvectors of Laplacian or signless Laplacian of Hypergraphs Associated with Zero Eigenvalue

Abstract

Let $G$ be a connected $m$-uniform hypergraph. In this paper we mainly consider the eigenvectors of the Laplacian or signless Laplacian tensor of $G$ associated with zero eigenvalue, called the first Laplacian or signless Laplacian eigenvectors of $G$. By means of the incidence matrix of $G$, the number of first Laplacian or signless Laplaican (H- or N-)eigenvectors can be get explicitly by solving the Smith normal form of the incidence matrix over ${\mathbb{Z}}_m$ or ${\mathbb{Z}}_2$. Consequently, we prove that the number of first Laplacian (H-)eigenvectors is equal to the number of first signless Laplacian (H-)eigenvectors when zero is an (H-)eigenvalue of the signless Laplacian tensor. We establish a connection between first Laplacian (signless Laplacian) H-eigenvectors and the even (odd) bipartitions of $G$.

References in corpus (9)

• Perron-Frobenius theorem for nonnegative multilinear forms and extensions
• The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph
• On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs
• On some properties of nonnegative weakly irreducible tensors
• The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs
• The $H$-spectrum of a generalized power hypergraph
• Eigenvariety of Nonnegative Symmetric Weakly Irreducible Tensors Associated with Spectral Radius and Its Application to Hypergraphs
• The largest $H$-eigenvalue and spectral radius of Laplacian tensor of non-odd-bipartite generalized power hypergraphs
• Some Spectral Properties of Odd-Bipartite $Z$-Tensors and Their Absolute Tensors

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• The least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices
• The least H-eigenvalue of signless Laplacian of non-odd-bipartite hypergraphs
• Minimal non-odd-transversal hypergraphs and minimal non-odd-bipartite hypergraphs