On the second-largest modulus among the eigenvalues of a power hypergraph
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915215245836288 |
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| author | Bu, Changjiang Chen, Lixiang Shi, Yongtang |
| author_facet | Bu, Changjiang Chen, Lixiang Shi, Yongtang |
| contents | It is well known that the algebraic multiplicity of an eigenvalue of a graph (or real symmetric matrix) is equal to the dimension of its corresponding linear eigen-subspace, also known as the geometric multiplicity. However, for hypergraphs, the relationship between these two multiplicities remains an open problem. For a graph $G=(V,E)$ and $k \geq 3$, the $k$-power hypergraph $G^{(k)}$ is a $k$-uniform hypergraph obtained by adding $k-2$ new vertices to each edge of $G$, who always has non-real eigenvalues. In this paper, we determine the second-largest modulus $Λ$ among the eigenvalues of $G^{(k)}$, which is indeed an eigenvalue of $G^{(k)}$. The projective eigenvariety $\mathbb{V}_Λ$ associated with $Λ$ is the set of the eigenvectors of $G^{(k)}$ corresponding to $Λ$ considered in the complex projective space. We show that the dimension of $\mathbb{V}_Λ$ is zero, i.e, there are finitely many eigenvectors corresponding to $Λ$ up to a scalar. We give both the algebraic multiplicity of $Λ$ and the total multiplicity of the eigenvector in $\mathbb{V}_Λ$ in terms of the number of the weakest edges of $G$. Our result show that these two multiplicities are equal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21174 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the second-largest modulus among the eigenvalues of a power hypergraph Bu, Changjiang Chen, Lixiang Shi, Yongtang Combinatorics 05C50, 05C65 It is well known that the algebraic multiplicity of an eigenvalue of a graph (or real symmetric matrix) is equal to the dimension of its corresponding linear eigen-subspace, also known as the geometric multiplicity. However, for hypergraphs, the relationship between these two multiplicities remains an open problem. For a graph $G=(V,E)$ and $k \geq 3$, the $k$-power hypergraph $G^{(k)}$ is a $k$-uniform hypergraph obtained by adding $k-2$ new vertices to each edge of $G$, who always has non-real eigenvalues. In this paper, we determine the second-largest modulus $Λ$ among the eigenvalues of $G^{(k)}$, which is indeed an eigenvalue of $G^{(k)}$. The projective eigenvariety $\mathbb{V}_Λ$ associated with $Λ$ is the set of the eigenvectors of $G^{(k)}$ corresponding to $Λ$ considered in the complex projective space. We show that the dimension of $\mathbb{V}_Λ$ is zero, i.e, there are finitely many eigenvectors corresponding to $Λ$ up to a scalar. We give both the algebraic multiplicity of $Λ$ and the total multiplicity of the eigenvector in $\mathbb{V}_Λ$ in terms of the number of the weakest edges of $G$. Our result show that these two multiplicities are equal. |
| title | On the second-largest modulus among the eigenvalues of a power hypergraph |
| topic | Combinatorics 05C50, 05C65 |
| url | https://arxiv.org/abs/2503.21174 |