Toric ideals and diagonal 2-minors

TL;DR

This paper explores the algebraic structure of ideals generated by diagonal 2-minors of matrices associated with graphs, characterizing when these ideals are toric and analyzing their initial ideals and Gr{"o}bner bases.

Contribution

It provides a complete characterization of graphs for which the associated ideals are toric and describes properties of their initial ideals and Gr{"o}bner bases.

Findings

Initial ideals of $P_G$ are generated by squarefree monomials with degree bounds for bipartite graphs.

Characterization of graphs where $P_G$ is a toric ideal.

Computed universal Gr{"o}bner bases in specific cases.

Abstract

Let $G$ be a simple graph on the vertex set $\{1,\ldots,n\}$ with $m$ edges. An algebraic object attached to $G$ is the ideal $P_{G}$ generated by diagonal 2-minors of an $n \times n$ matrix of variables. In this paper we prove that if $G$ is bipartite, then every initial ideal of $P_{G}$ is generated by squarefree monomials of degree at most $\left \lfloor{\frac{m+n+1}{2}} \right \rfloor$. Furthermore, we completely characterize all connected graphs $G$ for which $P_{G}$ is the toric ideal associated to a finite simple graph. Finally we compute in certain cases the universal Gr{\"o}bner basis of $P_{G}$.