Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 6

Special issue in honor of Fan Chung

Guest editors: Paul Horn, Yong Lin, and Linyuan Lu

Geometric vs algebraic nullity for hyperpaths

Pages: 2433 – 2460

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n6.a5

Authors

Joshua Cooper (Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)

Grant Fickes (Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)

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.

Received 3 July 2021

Received revised 4 March 2022

Accepted 13 March 2022

Published 29 March 2023