Multidegrees of binomial edge ideals

TL;DR

This paper develops a combinatorial method to compute the multidegree of binomial edge ideals of graphs, linking algebraic invariants to graph properties, and explicitly calculates these for various special graph classes.

Contribution

It introduces a way to determine the multidegree of binomial edge ideals using combinatorial properties and computes these for specific graph families.

Findings

Provides a formula for multidegree based on $S_{min}(G)$

Calculates $S_{min}(G)$ for star, cycle, and other graphs

Determines multidegrees for several graph classes

Abstract

Let $G$ be a simple graph with binomial edge ideal $J_G$. We prove how to calculate the multidegree of $J_G$ based on combinatorial properties of $G$. In particular, we study the set $S_{\min}(G)$ defined as the collection of subsets of vertices whose prime ideals have minimum codimension. We provide results which assist in determining $S_{\min}(G)$, then calculate $S_{\min}(G)$ for star, horned complete, barbell, cycle, wheel, and friendship graphs, and use the main result of the paper to obtain the multidegrees of their binomial edge ideals.