Toric ideals of normalized graph algebras

TL;DR

This paper introduces a graph-theoretic approach to characterize the minimal generators of the integral closure of quadratic monomial algebras and describes their toric ideals using graphical binomials.

Contribution

It presents a simpler, graph-based method to determine generators and relations of the integral closure of quadratic monomial algebras, improving on existing techniques.

Findings

The minimal generators are characterized using a new graph-theoretic method.

The toric ideal of relations is generated by graphically defined binomials.

The spectra of the original algebra and its integral closure are homeomorphic.

Abstract

A graph-theoretic method, simpler than existing ones, is used to characterize the minimal set of monomial generators for the integral closure of any algebra of polynomials generated by quadratic monomials. The toric ideal of relations between these generators is generated by a set of binomials, defined graphically. The spectra of the original algebra and of its integral closure turn out to be canonically homeomorphic.