Graphs with few trivial critical ideals

Alfaro, Carlos A.; Valencia, Carlos E.

doi:10.1016/j.endm.2015.07.065

Cited by 4 publications

(2 citation statements)

References 22 publications

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“…For instance, in [2], critical ideals were used to classify the simple graphs whose critical group has two of their invariant factors equal to one. Also, in [3] and [4], they were used to classify the simple graphs whose critical group has three of their invariant factors equal to one and bounded clique number.…”

Section: Introductionmentioning

confidence: 99%

On the critical ideals of graphs

Corrales

Valencia

2013

Linear Algebra and its Applications

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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.Comment: 23 pages. Some changes over the previous version. Accepted in Linear algebra and its Application

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Section: Introductionmentioning

confidence: 99%

On the critical ideals of graphs

Corrales

Valencia

2013

Linear Algebra and its Applications

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“…For i ≥ 3, the set Forb(Γ ≤i ) is more complex than Forb(Γ ≤1 ) and Forb(Γ ≤2 ). For instance, in [1] was proved that Forb(Γ ≤3 ) has 49 graphs. Moreover, we conjecture that Forb(Γ ≤k ) is finite.…”

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Graphs with two trivial critical ideals

Alfaro

Valencia

2014

Discrete Applied Mathematics

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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 ≤k , are closed under induced subgraphs. In this article we find the set of minimal forbidden subgraphs for Γ ≤2 , and we use this forbidden subgraphs to get a classification of the graphs in Γ ≤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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A note on the critical ideals of a cycle

Chen

2022

Discrete Mathematics

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Graphs with few trivial critical ideals

Cited by 4 publications

References 22 publications

On the critical ideals of graphs

On the critical ideals of graphs

Graphs with two trivial critical ideals

A note on the critical ideals of a cycle

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