New linear invariants of hypergraphs

TL;DR

This paper introduces a new family of linear invariants for hypergraphs, defining equivalence relations and quotient hypergraphs that reveal structural properties and include a universal invariant with a closure property.

Contribution

It presents a novel parameterized family of linear invariants for hypergraphs, including a universal invariant and a closure operation, advancing the understanding of hypergraph structure.

Findings

The invariant $ ext{Sig}(ullet, T)$ embeds into $ ext{Sig}(ullet, U)$, showing universality.

The $U$-fusion operation acts as a closure, simplifying hypergraphs in a one-time process.

$U$-fusion refines all other $T$-fusions, establishing a hierarchy of invariants.

Abstract

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 $U(\lambda)=\lambda(1)+\cdots+\lambda(\ell)$, 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.