On the critical ideals of graphs

TL;DR

This paper introduces critical ideals of digraphs, generalizing several graph invariants, and determines their algebraic structure for specific graph classes, linking them to graph stability and clique numbers.

Contribution

It defines critical ideals of digraphs, finds minimal generators and Grobner bases for complete graphs, cycles, and paths, and relates critical ideals to graph stability and clique numbers.

Findings

Determined minimal generators for critical ideals of specific graphs.

Established Grobner bases for these critical ideals.

Linked the number of trivial critical ideals to stability and clique numbers.

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.