Edge Ideals of Weighted Graphs
Commutative Algebra
2013-02-26 v2 Combinatorics
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.
Cite
@article{arxiv.1205.3678,
title = {Edge Ideals of Weighted Graphs},
author = {Chelsey Paulsen and Sean Sather-Wagstaff},
journal= {arXiv preprint arXiv:1205.3678},
year = {2013}
}
Comments
20 pages, uses xypic, v.2 includes new Remark 3.7, Theorem 5.7, and Remark 5.8; and the proof of Theorem 5.10 has been modified