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 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.
Original language | English |
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Pages (from-to) | 3870-3892 |
Number of pages | 23 |
Journal | Linear Algebra and Its Applications |
Volume | 439 |
Issue number | 12 |
DOIs | |
State | Published - 15 Dec 2013 |
Externally published | Yes |
Keywords
- Clique number
- Complete graph
- Critical group
- Critical ideals
- Cycle
- Determinantal ideal
- Generalized Laplacian matrix
- Gröbner basis
- Stability number