A hypergraph Heilmann--Lieb theorem
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866912299593236480 |
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| author | Wan, Jiang-Chao Wang, Yi Fan, Yi-zheng |
| author_facet | Wan, Jiang-Chao Wang, Yi Fan, Yi-zheng |
| contents | The Heilmann--Lieb theorem is a fundamental theorem in algebraic combinatorics which provides a characterization of the distribution of the zeros of matching polynomials of graphs. In this paper, we establish a hypergraph Heilmann--Lieb theorem as follows. Let $\h$ be a connected $k$-graph with maximum degree $Δ\geq 2$ and let $μ(\h, x)$ be its matching polynomial. We show that the zeros (with multiplicities) of $μ(\h, x)$ are invariant under a rotation of an angle $2π/{\ell}$ in the complex plane for some positive integer $\ell$ and $k$ is the maximum integer with this property. We further prove that the maximum modulus $λ(\h)$ of all the zeros of $μ(\h, x)$ is a simple root of $μ(\h, x)$ and satisfies $$Δ^{\frac{1}{ k}} \leq λ(\h)< \frac{k}{k-1}\big((k-1)(Δ-1)\big)^{\frac{1}{ k}}.$$ To achieve these, we prove that $μ(\h, x)$ divides the matching polynomial of the $k$-walk-tree of $\h$, which generalizes a classical result due to Godsil from graphs to hypergraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_09558 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A hypergraph Heilmann--Lieb theorem Wan, Jiang-Chao Wang, Yi Fan, Yi-zheng Combinatorics The Heilmann--Lieb theorem is a fundamental theorem in algebraic combinatorics which provides a characterization of the distribution of the zeros of matching polynomials of graphs. In this paper, we establish a hypergraph Heilmann--Lieb theorem as follows. Let $\h$ be a connected $k$-graph with maximum degree $Δ\geq 2$ and let $μ(\h, x)$ be its matching polynomial. We show that the zeros (with multiplicities) of $μ(\h, x)$ are invariant under a rotation of an angle $2π/{\ell}$ in the complex plane for some positive integer $\ell$ and $k$ is the maximum integer with this property. We further prove that the maximum modulus $λ(\h)$ of all the zeros of $μ(\h, x)$ is a simple root of $μ(\h, x)$ and satisfies $$Δ^{\frac{1}{ k}} \leq λ(\h)< \frac{k}{k-1}\big((k-1)(Δ-1)\big)^{\frac{1}{ k}}.$$ To achieve these, we prove that $μ(\h, x)$ divides the matching polynomial of the $k$-walk-tree of $\h$, which generalizes a classical result due to Godsil from graphs to hypergraphs. |
| title | A hypergraph Heilmann--Lieb theorem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2206.09558 |