Graphs with few trivial critical ideals

doi:10.1016/j.endm.2015.07.065

Electronic Notes in Discrete Mathematics

Volume 50, December 2015, Pages 391-396

https://doi.org/10.1016/j.endm.2015.07.065 Get rights and content

Abstract

The critical ideals of a graph are determinantal ideals of the generalized Laplacian matrix associated to a graph. Let $Γ_{\leq i}$ denote the set of simple connected graphs with at most i trivial critical ideals. The main goal is to obtain a characterization of the graphs in $Γ_{\leq 3}$ with clique number equal to 2, and the graphs in $Γ_{\leq 3}$ with clique number equal to 3. This shows that there exists a strong connection between the structural properties of the graph (like the clique number and the stability number) with its critical ideals.

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Cited by (2)

A note on the critical ideals of a cycle
2022, Discrete Mathematics
Citation Excerpt :
More precisely, we get that the critical group of G from its critical ideals via some specialization, see [7, Proposition 3.6]. Besides the above, the critical ideals are also related to other combinatoric properties of G, such as, the clique number, the zero forcing number and the algebraic co-rank, see [7,1–4]. Until now, critical ideals have only been computed for a few classes of graphs.
Critical ideals of a graph G which are the determinantal ideals of its generalized Laplacian matrix were first introduced by Corrales and Valencia as a generalization of the critical group. Then it was shown that critical ideals are also closely related to other properties of the graph, such as the clique number and the zero forcing number. In this note, we give a simple proof of Theorem 4.13 proved in [7], which gives a Gröbner basis of the first nontrivial critical ideal of a cycle. After that as applications we determine explicit expressions for the characteristic ideals of a cycle and the critical groups of a family of thick wheels.
Digraphs with at most one trivial critical ideal
2018, Linear and Multilinear Algebra

¹: Carlos A. Alfaro was supported by CONACyT grant 166059 and Carlos E. Valencia was supported by SNI and CONACyT grant 166059.

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Graphs with few trivial critical ideals

Abstract

Linear Algebra and Its Applications

Discrete Applied Mathematics

Linear Algebra and its Applications

J. Combin. Theory B

Discrete Math.

Linear Algebra Appl.

Small clique number graphs with three trivial critical ideals

The lattice of integral flows and the lattice of integral cuts on a finite graph

Bull. Soc. Math. France