Geometric vs Algebraic Nullity for Hyperpaths

TL;DR

This paper investigates the relationship between geometric and algebraic nullities in hypergraphs, specifically analyzing the zero-eigenvariety of a 3-hyperpath and verifying a conjecture linking geometric dimension and nullity.

Contribution

It fully describes the zero-eigenvariety of a 3-hyperpath, computes its algebraic nullity, and verifies a conjecture relating geometric and algebraic nullities for this class.

Findings

Complete description of the zero-eigenvariety of a 3-hyperpath

Calculation of the algebraic nullity for this hypergraph class

Verification of the conjecture relating geometric dimension and nullity

Abstract

We consider the question of how the eigenvarieties of a hypergraph relate to the algebraic multiplicities of their corresponding eigenvalues. Specifically, we (1) fully describe the irreducible components of the zero-eigenvariety of a loose $3$-hyperpath (its "nullvariety"), (2) use recent results of Bao-Fan-Wang-Zhu to compute the corresponding algebraic multiplicity of zero (its "nullity"), and then (3) for this special class of hypergraphs, verify a conjecture of Hu-Ye about the relationship between the geometric (multi-)dimension of the nullvariety and the nullity.