New linear invariants of hypergraphs

TL;DR

The paper develops a linear-invariant framework for ell-uniform hypergraphs by introducing $T$-signals and the corresponding fusion operations. It proves a universal property for the map $U$, showing that $U$-fusion yields the most refined fusion, and defines the $U$-frame as a closure operator that computably simplifies hypergraphs in polynomial time. Through theoretical results and algorithmic constructions, it demonstrates how frames capture essential structural features while remaining efficiently computable, and provides empirical insights into the level of simplification achievable in practice. The work also discusses limitations compared to full tensor derivations and points to rich avenues for extending hypergraph invariants using the broader Strata framework.

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 $U(λ)=λ(1)+\cdots+λ(\ell)$, 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.

Figures (9)

• Figure 1: A hypergraph on 5 vertices with 3 hyperedges.
• Figure 2: Quotients of $\mathcal{H}$ by two different equivalence relations.
• Figure 3: On the left is a 3-uniform hypergaph $\mathcal{H}$ on 5 vertices. On the right is the $5\times 5\times 5$ array $\Gamma$ corresponding to $\mathcal{H}$, where the $3!=6$ nonzero entries for each colored edge are represented as blocks of the same color. This helps to see the symmetry of $\Gamma$.
• Figure 4: On the left is our hypergraph $\mathcal{H}$ on 5 vertices with 3 hyperedges, and on the right is its derivation quotient $\mathop{\mathrm{\mathfrak{F}}}\nolimits(\mathcal{H})$.
• Figure 5: A hypergraph $\mathcal{H}$ on 15 vertices with 10 hyperedges.

Theorems & Definitions (35)

• Definition 2.1: Hypergraph
• Definition 2.2: Connectedness
• Definition 2.3: Quotients
• Example 2.4
• Example 2.5
• Remark 3.1
• Definition 3.2: $T$-Signal
• Example 3.3
• Lemma 3.4
• proof