On the critical ideals of graphs
Commutative Algebra
2014-02-21 v3 Combinatorics
Abstract
We introduce some determinantal ideals of the generalized Laplacian matrix associated to a digraph G, that we call critical ideals of G. Critical ideals generalize the critical group and the characteristic polynomials of the adjacency and Laplacian matrices of a digraph. The main results of this article are the determination of some minimal generator sets and the reduced Grobner basis for the critical ideals of the complete graphs, the cycles and the paths. Also, we establish a bound between the number of trivial critical ideals and the stability and clique numbers of a graph.
Keywords
Cite
@article{arxiv.1205.3105,
title = {On the critical ideals of graphs},
author = {Hugo Corrales and Carlos E. Valencia},
journal= {arXiv preprint arXiv:1205.3105},
year = {2014}
}
Comments
23 pages. Some changes over the previous version. Accepted in Linear algebra and its Applications