Edge Ideals of Weighted Graphs

TL;DR

This paper explores the algebraic properties of edge ideals derived from weighted graphs, providing combinatorial characterizations of their decompositions and conditions for Cohen-Macaulayness.

Contribution

It introduces combinatorial descriptions of m-irreducible decompositions for weighted graph edge ideals and characterizes when these ideals are unmixed or Cohen-Macaulay.

Findings

Weighted complete graphs are Cohen-Macaulay.

Characterizations of unmixed weighted cycles, suspensions, and trees.

Descriptions of m-irreducible decompositions via weighted vertex covers.

Abstract

We study weighted graphs and their "edge ideals" which are ideals in polynomial rings that are defined in terms of the graphs. We provide combinatorial descriptions of m-irreducible decompositions for the edge ideal of a weighted graph in terms of the combinatorics of "weighted vertex covers". We use these, for instance, to say when these ideals are m-unmixed. We explicitly describe which weighted cycles, suspensions, and trees are unmixed and which ones are Cohen-Macaulay, and we prove that all weighted complete graphs are Cohen-Macaulay.