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arXiv · 1807.00544

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

Yi-Zheng Fan·Yi Wang·Yan-Hong Bao·Jiang-Chao Wan·Min Li·Zhu Zhu
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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 $G$.

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Yi-Zheng Fan, Yi Wang, Yan-Hong Bao, Jiang-Chao Wan, Min Li, Zhu Zhu. 2019-01-23. Eigenvectors of Laplacian or signless Laplacian of Hypergraphs Associated with Zero Eigenvalue. https://doi.org/10.1016/j.laa.2019.06.001

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