Persistent commutative algebra on graphs and hypergraphs
Abstract
We introduce a persistent commutative algebra for studying the algebraic and combinatorial evolution of edge ideals of graphs and hypergraphs under filtration. Building on the Persistent Stanley--Reisner Theory (PSRT), we develop the notion of persistent edge ideals and analyze their graded Betti numbers across the filtration of graphs or hypergraphs. To enable this analysis, we establish a persistent extension of Hochster's formula, providing a functorial correspondence between algebraic and topological persistence. We further examine the behavior of Betti splittings in the persistent setting, proving a general inequality that extends the classical splitting result to the filtration of monomial ideals. Motivated by graph-theoretic interpretations, we introduce persistent minimal vertex covers, which encode the temporal structure of combinatorial dependencies within evolving graphs or hypergraphs. Applications to alignment-free genomic classification and molecular isomer discrimination demonstrate the interpretability and representatbility of persistent edge ideals as algebraic invariants, bridging combinatorial commutative algebra and data science.
Cite
@article{arxiv.2512.17619,
title = {Persistent commutative algebra on graphs and hypergraphs},
author = {Faisal Suwayyid and Guo-Wei Wei},
journal= {arXiv preprint arXiv:2512.17619},
year = {2025}
}