Graphs with two trivial critical ideals

Abstract

The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. A basic property of the critical ideals of graphs asserts that the graphs with at most k trivial critical ideals, ${\Gamma }_{\le k}$, are closed under induced subgraphs. In this article we find the set of minimal forbidden subgraphs for ${\Gamma }_{\le 2}$, and we use this forbidden subgraphs to get a classification of the graphs in ${\Gamma }_{\le 2}$. As a consequence we give a classification of the simple graphs whose critical group has two invariant factors equal to one. At the end of this article we give two infinite families of forbidden subgraphs.

15 pages, 2 figures

References in corpus (2)

• On the critical ideals of graphs
• A note on the third invariant factor of the Laplacian matrix of a graph

Cited by in corpus (6)

• On the critical ideals of graphs
• Graph classes for critical ideals, minimum rank and zero forcing number
• Eigenvalues, Smith normal form and determinantal ideals
• Critical ideals of trees
• A Survey of directed graphs invariants
• Digraphs with at most one trivial critical ideal