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arXiv · 1304.4211

Graphs with two trivial critical ideals

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

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Carlos A. Alfaro, Carlos E. Valencia. 2013-04-15. Graphs with two trivial critical ideals. https://doi.org/10.1016/j.dam.2013.11.002

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