Toric Ideals of Weighted Oriented Graphs

TL;DR

This paper studies the algebraic structure of toric ideals associated with vertex-weighted oriented graphs, providing generators, conditions for triviality, and characterizations of binomial generation based on graph structure.

Contribution

It introduces new results on the generators of toric ideals for weighted oriented graphs, extending known results from unweighted, unoriented graphs.

Findings

Identifies generating sets for toric ideals in certain graph classes.

Characterizes when the toric ideal is trivial (zero ideal).

Provides conditions for the toric ideal to be generated by a single binomial.

Abstract

Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result which is analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in term of the structure of the graph.