Characteristic Polynomials and Hypergraph Generating Functions via Heaps of Pieces

TL;DR

This paper extends classical graph generating function results to hypergraphs using Heaps of Pieces, linking characteristic polynomials to combinatorial objects in hypergraphs through algebraic and combinatorial methods.

Contribution

It introduces a hypergraph analogue of the classical generating function relation using Heaps of Pieces and algebraic resultants, broadening combinatorial and algebraic understanding.

Findings

Hypergraph generating functions relate to combinatorial structures called infragraphs.

Heaps of Pieces framework applies to hypergraphs, graphs, and digraphs.

Multivariate resultants of polynomial systems are used to analyze hypergraph properties.

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 \[ \frac{\phi_{G-u}(t)}{\phi_G(t)} \] where $\phi_G(t)$ 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.