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TitleOn the critical ideals of graphs
Author(s) Hugo Corrales, Carlos E. Valencia
TypeArticle in Journal
AbstractAbstract 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 Gröbner 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.
KeywordsCritical ideals, Critical group, General Clique number, Cycle, Complete graph
URL http://www.sciencedirect.com/science/article/pii/S0024379513006307
JournalLinear Algebra and its Applications
Pages3870 - 3892
Translation No
Refereed No