Oriented hypergraphs and generalizing the Harary-Sachs theorem to integer matrices

TL;DR

This paper extends the Harary-Sachs theorem to integer matrices using oriented hypergraphs, introducing contributors and tail-equivalence to analyze minors and spectral properties of Laplacians.

Contribution

It generalizes cycle cover concepts to contributors via oriented hypergraphs, providing new combinatorial tools for analyzing Laplacian minors and spectra.

Findings

Characterization of minors using contributors and hypergraphs

Cancellative property of tail-equivalence grouping

Proof of non-0 isospectrality of Laplacian polynomials

Abstract

Incidence-based generalizations of cycle covers, called contributors, extend the Harary-Sachs coefficient theorem for characteristic polynomials of the adjacency matrix of graphs. All minors of the Laplacian resulting from an integer matrix are characterized using their associated oriented hypergraph through a new minimal collection of contributors to produce the coefficients of the total-minor polynomial. We prove that the natural grouping of contributors via tail-equivalence is necessarily cancellative for any contributor family that reuses an edge. We then provide a new combinatorial proof on the non-0 isospectrality of the traditional characteristic polynomials of the Laplacian and its dual.