Graphs with two trivial critical ideals

TL;DR

This paper characterizes graphs with at most two trivial critical ideals by identifying minimal forbidden subgraphs and classifying such graphs, linking algebraic properties to graph structure.

Contribution

It finds the minimal forbidden subgraphs for graphs with at most two trivial critical ideals and classifies these graphs, connecting algebraic invariants to graph structure.

Findings

Identified minimal forbidden subgraphs for $ ext{Gamma}_{ ext{leq } 2}$.

Classified graphs with at most two trivial critical ideals.

Provided two infinite families of forbidden subgraphs.

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, $\Gamma_{\leq k}$, are closed under induced subgraphs. In this article we find the set of minimal forbidden subgraphs for $\Gamma_{\leq 2}$, and we use this forbidden subgraphs to get a classification of the graphs in $\Gamma_{\leq 2}$. 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.