The Principal Component of the Jets of a Graph

TL;DR

This paper introduces the s-order principal component of the jets of a graph, linking algebraic properties of edge ideals to graph covers, with applications to cochordal graphs and computational methods.

Contribution

It defines the s-order principal component of jets of a graph and relates its primary decomposition to minimal vertex covers, extending to cochordal graphs and Froberg's theorem.

Findings

The s-order principal component of jets of a cochordal graph is cochordal.

Provides a description of the primary decomposition of edge ideals in terms of minimal vertex covers.

Includes computational methods using Macaulay2 for jet calculations.

Abstract

We define the s-order principal component of the jets of a graph and give a description of the primary decomposition of its edge ideal in terms of the minimal vertex covers of the base graph. As an application, we show the s-order principal component of the jets of a cochordal graph is cochordal, and connect this to Froberg's theorem on the linear resolution of edge ideals of cochordal graphs. An appendix is provided describing some computations of jets in the computer algebra system Macaulay2.