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Characteristic Polynomials and Hypergraph Generating Functions via Heaps of Pieces

Joshua Cooper, Krystal Guo, Utku Okur

TL;DR

This work extends Jacobi's classical relation between closed walks and the graph's characteristic polynomials to hypergraphs by leveraging Viennot's Heaps of Pieces. It develops a comprehensive framework in which infragraphs serve as the fundamental pieces, and cycles plus edgegons (or infragraphs) form the building blocks whose concatenations (pyramids) encode the combinatorics of walks and coefficients. Central results express normalized characteristic polynomials and their logarithms as sums over trivial heaps and pyramids, yielding Harary–Sachs-type formulas for hypergraphs, including root and edge-variable variants. The approach unifies counting of infragraph embeddings, rootings, and arborescences via Kocay's lemma and BEST-type theorems, with multiplicativity under disjoint unions enabling practical computations and paving the way for hypergraph spectral theory extensions.

Abstract

It is a classical result due to Jacobi in algebraic combinatorics that the generating function of closed walks at a vertex $u$ in a graph $G$ is determined by the rational function \[ \frac{φ_{G-u}(t)}{φ_G(t)} \] where $φ_G(t)$ is the characteristic polynomial of $G$. In this paper, we show that the corresponding rational function for a hypergraph is also a generating function for some combinatorial objects in the hypergraph. We make use of the Heaps of Pieces framework, developed by Viennot, demonstrating its use on graphs, digraphs, and multigraphs before using it on hypergraphs. In the case of a graph $G$, the pieces are cycles and the concurrence relation is sharing a vertex. The pyramids with maximal piece containing a vertex $u \in V(G)$ are in one-to-one correspondence with closed walks at $u$. In the case of a hypergraph $\mathcal{H}$, connected "infragraphs" can be defined as the set of pieces, with the same concurrence relation: sharing a vertex. Our main results are established by analyzing multivariate resultants of polynomial systems associated to adjacency hypermatrices.

Characteristic Polynomials and Hypergraph Generating Functions via Heaps of Pieces

TL;DR

This work extends Jacobi's classical relation between closed walks and the graph's characteristic polynomials to hypergraphs by leveraging Viennot's Heaps of Pieces. It develops a comprehensive framework in which infragraphs serve as the fundamental pieces, and cycles plus edgegons (or infragraphs) form the building blocks whose concatenations (pyramids) encode the combinatorics of walks and coefficients. Central results express normalized characteristic polynomials and their logarithms as sums over trivial heaps and pyramids, yielding Harary–Sachs-type formulas for hypergraphs, including root and edge-variable variants. The approach unifies counting of infragraph embeddings, rootings, and arborescences via Kocay's lemma and BEST-type theorems, with multiplicativity under disjoint unions enabling practical computations and paving the way for hypergraph spectral theory extensions.

Abstract

It is a classical result due to Jacobi in algebraic combinatorics that the generating function of closed walks at a vertex in a graph is determined by the rational function where is the characteristic polynomial of . In this paper, we show that the corresponding rational function for a hypergraph is also a generating function for some combinatorial objects in the hypergraph. We make use of the Heaps of Pieces framework, developed by Viennot, demonstrating its use on graphs, digraphs, and multigraphs before using it on hypergraphs. In the case of a graph , the pieces are cycles and the concurrence relation is sharing a vertex. The pyramids with maximal piece containing a vertex are in one-to-one correspondence with closed walks at . In the case of a hypergraph , connected "infragraphs" can be defined as the set of pieces, with the same concurrence relation: sharing a vertex. Our main results are established by analyzing multivariate resultants of polynomial systems associated to adjacency hypermatrices.

Paper Structure

This paper contains 21 sections, 27 theorems, 141 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Heaps of Pieces and the Characteristic Polynomial of a Graph
  3. Preliminaries
  4. Operations on the Trails of an Eulerian Digraph
  5. The Heaps of Pieces Framework
  6. Bijections between Pyramids and Trails/Walks
  7. The Characteristic Polynomial of a Graph as a sum over Trivial Heaps
  8. An Example: Heaps of Pieces of a Simple Graph
  9. Heaps of Pieces on Hypergraphs
  10. Preliminaries
  11. Number of copies of an infragraph in a simple hypergraph
  12. Rooted stars and Rootings
  13. The Associated Coefficient
  14. The Harary-Sachs Theorem for Hypergraphs
  15. Kocay's Lemma and Alternative Expression of the Characteristic Polynomial
  16. ...and 6 more sections

Key Result

Lemma 2.1

Given a heap $H = ( \Omega,\leq, \ell )$ and an element $\omega\in \Omega$, then the down-set $\downarrow \! \{ \omega \}_{H}$ is a pyramid. Let $H_1:=( \downarrow \! \{ \omega \}_{H} , \leq_{H_1}, \ell_{H_1} )$ and $H_2=( H\setminus \downarrow \! \{ \omega \}_{H} , \leq_{H_2}, \ell_{H_2} )$ be

Figures (7)

  • Figure 1: A digraph $D$
  • Figure 3: A digraph $D$ and some of its simple closed trails.
  • Figure 4: An example of a closed trail.
  • Figure 5: The host hypergraph $\mathcal{H}$ and $G \in \mathcal{V}^{1}_9$, a connected Veblen hypergraph on 9 edges.
  • Figure 6: The simple host hypergraph $\mathcal{H}$ and its infragraph $X \in \text{Inf}( \mathcal{H} )$
  • ...and 2 more figures

Theorems & Definitions (82)

  • Definition 1: Vertex-fixing isomorphisms of multi-graphs
  • Definition 2: Flattening of a multi-hypergraph
  • Definition 3
  • Remark 1
  • Remark 2
  • Example 1
  • Definition 4
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 72 more