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arXiv · 1901.08222

Eigenvectors of Z-tensors associated with least H-eigenvalue with application to hypergraphs

Yi-Zheng Fan·Yi Wang·Yan-Hong Bao
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Abstract

Unlike an irreducible $Z$-matrices, a weakly irreducible $Z$-tensor $\mathcal{A}$ can have more than one eigenvector associated with the least H-eigenvalue. We show that there are finitely many eigenvectors of $\mathcal{A}$ associated with the least H-eigenvalue. If $\mathcal{A}$ is further combinatorial symmetric, the number of such eigenvectors can be obtained explicitly by the Smith normal form of the incidence matrix of $\mathcal{A}$. When applying to a connected uniform hypergraph $G$, we prove that the number of Laplacian eigenvectors of $G$ associated with the zero eigenvalue is equal to the the number of adjacency eigenvectors of $G$ associated with the spectral radius, which is also equal to the number of signless Laplacian eigenvectors of $G$ associated with the zero eigenvalue if zero is an signless Laplacian eigenvalue.

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Yi-Zheng Fan, Yi Wang, Yan-Hong Bao. 2019-01-24. Eigenvectors of Z-tensors associated with least H-eigenvalue with application to hypergraphs. https://arxiv.org/abs/1901.08222

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