On the critical ideals of graphs

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.

23 pages. Some changes over the previous version. Accepted in Linear algebra and its Applications

References in corpus (1)

• Graphs with two trivial critical ideals

Cited by in corpus (5)

• Graphs with two trivial critical ideals
• Graph classes for critical ideals, minimum rank and zero forcing number
• Eigenvalues, Smith normal form and determinantal ideals
• Critical ideals of trees
• Degree and algebraic properties of lattice and matrix ideals