New linear invariants of hypergraphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917121416495104 |
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| author | Brooksbank, Peter A. Chaplin, Clara R. |
| author_facet | Brooksbank, Peter A. Chaplin, Clara R. |
| contents | We introduce a parameterized family of invariants for $\ell$-uniform hypergraphs. To each $\mathbb{K}$-linear transformation $T:\mathbb{K}^{\ell}\to \mathbb{K}^r$ we associate a function $\mathrm{Sig}(-,T)$ that maps $\ell$-uniform hypergraphs to $\mathbb{K}$-vector spaces. Given an $\ell$-uniform hypergraph $\mathcal{H}=(V,E)$, we use $\mathrm{Sig}(\mathcal{H},T)$ to define an equivalence relation $\equiv_T$ on $V$ called $T$-fusion, which determines a quotient hypergraph $\mathfrak{F}(\mathcal{H},T)$ called the $T$-frame of $\mathcal{H}$. We show that the map $U:\mathbb{K}^{\ell}\to \mathbb{K}$, where $U(λ)=λ(1)+\cdots+λ(\ell)$, is universal in that $\mathrm{Sig}(\mathcal{H},T)$ embeds in $\mathrm{Sig}(\mathcal{H},U)$, and $U$-fusion refines $T$-fusion for any $T:\mathbb{K}^{\ell}\to\mathbb{K}^r$. We further show that $\mathfrak{F}(\mathfrak{F}(\mathcal{H},U),U)=\mathfrak{F}(\mathcal{H},U)$ for any $\ell$-uniform hypergraph $\mathcal{H}$, so $\mathfrak{F}(-,U)$ is a closure function on the set of $\ell$-uniform hypergraphs. We explore the properties of this one-time simplification of a hypergraph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03342 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New linear invariants of hypergraphs Brooksbank, Peter A. Chaplin, Clara R. Combinatorics 05C65, 15A69, 15-04, 15B30 We introduce a parameterized family of invariants for $\ell$-uniform hypergraphs. To each $\mathbb{K}$-linear transformation $T:\mathbb{K}^{\ell}\to \mathbb{K}^r$ we associate a function $\mathrm{Sig}(-,T)$ that maps $\ell$-uniform hypergraphs to $\mathbb{K}$-vector spaces. Given an $\ell$-uniform hypergraph $\mathcal{H}=(V,E)$, we use $\mathrm{Sig}(\mathcal{H},T)$ to define an equivalence relation $\equiv_T$ on $V$ called $T$-fusion, which determines a quotient hypergraph $\mathfrak{F}(\mathcal{H},T)$ called the $T$-frame of $\mathcal{H}$. We show that the map $U:\mathbb{K}^{\ell}\to \mathbb{K}$, where $U(λ)=λ(1)+\cdots+λ(\ell)$, is universal in that $\mathrm{Sig}(\mathcal{H},T)$ embeds in $\mathrm{Sig}(\mathcal{H},U)$, and $U$-fusion refines $T$-fusion for any $T:\mathbb{K}^{\ell}\to\mathbb{K}^r$. We further show that $\mathfrak{F}(\mathfrak{F}(\mathcal{H},U),U)=\mathfrak{F}(\mathcal{H},U)$ for any $\ell$-uniform hypergraph $\mathcal{H}$, so $\mathfrak{F}(-,U)$ is a closure function on the set of $\ell$-uniform hypergraphs. We explore the properties of this one-time simplification of a hypergraph. |
| title | New linear invariants of hypergraphs |
| topic | Combinatorics 05C65, 15A69, 15-04, 15B30 |
| url | https://arxiv.org/abs/2512.03342 |