Small clique number graphs with three trivial critical ideals

Alfaro, Carlos A.; Valencia, Carlos E.

doi:10.1515/spma-2018-0011

Cited by 11 publications

(15 citation statements)

References 20 publications

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“…Then Γ ≤1 = T * (−1) (K 1 ), where T * δ (G) denotes the set of induced subgraphs of one graph in T δ (G). As we mentioned before, in [3] it was proved that [2] it was given a similar result about Γ ≤3 . In general given a fixed constant k, we expect that Γ ≤k has a similar classification.…”

Section: An Upper Bound For the Algebraic Co-rank Of Graphs With Twinssupporting

confidence: 57%

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Critical ideals of signed graphs with twin vertices

Alfaro

Corrales²,

Valencia

2017

Advances in Applied Mathematics

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Abstract. This paper studies critical ideals of graphs with twin vertices, which are vertices with the same neighbors. A pair of such vertices are called replicated if they are adjacent, and duplicated, otherwise. Critical ideals of graphs having twin vertices have good properties and show regular patterns. Given a graph G = (V, E) and d ∈ Z |V | , let G d be the graph obtained from G by duplicating dv times or replicating −dv times the vertex v when dv > 0 or dv < 0, respectively. Moreover, given δ ∈ {0, 1, −1} |V | , let|V | such that dv = 0 if and only if δv = 0 and dvδv > 0 otherwise} be the set of graphs sharing the same pattern of duplication or replication of vertices. More than one half of the critical ideals of a graph in T δ (G) can be determined by the critical ideals of G. The algebraic co-rank of a graph G is the maximum integer i such that the i-th critical ideal of G is trivial. We show that the algebraic co-rank of any graph in T δ (G) is equal to the algebraic co-rank of G δ . Moreover, the algebraic co-rank can be determined by a simple evaluation of the critical ideals of G. For a large enough d ∈ Z V (G) , we show that the critical ideals of G d have similar behavior to the critical ideals of the disjoint union of G and some set {Kn v } {v∈V (G) | dv <0} of complete graphs and some set {Tn v } {v∈V (G) | dv >0} of trivial graphs. Additionally, we pose important conjectures on the distribution of the algebraic co-rank of the graphs with twins vertices. These conjectures imply that twin-free graphs have a large algebraic co-rank, meanwhile a graph having small algebraic co-rank has at least one pair of twin vertices.

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Section: An Upper Bound For the Algebraic Co-rank Of Graphs With Twinssupporting

confidence: 57%

“…In fact, this bound is tight since the equality holds for the complete graphs (see Example 2.10). This upper bound can be used in the classification of the graphs that have algebraic co-rank less than or equal to an integer k, see [3] and [2].…”

Section: An Upper Bound For the Algebraic Co-rank Of Graphs With Twinsmentioning

confidence: 99%

Critical ideals of signed graphs with twin vertices

Alfaro

Corrales²,

Valencia

2017

Advances in Applied Mathematics

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“…. These ideals were defined in [10] and further studied in [1,6,2,4,5], from which our study was originally inspired.…”

Section: 3mentioning

confidence: 99%

Distance ideals of graphs

Alfaro

Taylor

2020

Linear Algebra and its Applications

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“…These were called critical ideals, see [14]. They have been explored in [2,6,7,8], and in [1,4] it was found new connections in contexts different from the Smith group or Sandpile group like the zero-forcing number and the minimum rank of a graph. In this setting, the set of forbidden graphs for the family with at most k trivial critical ideals is conjectured to be finite, see [6,Conjecture 5.5].…”

Section: Lemma 4 [3]mentioning

confidence: 99%

On graphs with 2 trivial distance ideals

Alfaro

2020

Linear Algebra and its Applications

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Distance ideals generalize the Smith normal form of the distance matrix of a graph. The family of graphs with 2 trivial distance ideals contains the family of graphs whose distance matrix has at most 2 invariant factors equal to 1. Here we give an infinite family of forbidden induced subgraphs for the graphs with 2 trivial distance ideals. These are also related with other well known graph classes.

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Small clique number graphs with three trivial critical ideals

Cited by 11 publications

References 20 publications

Critical ideals of signed graphs with twin vertices

Critical ideals of signed graphs with twin vertices

Distance ideals of graphs

On graphs with 2 trivial distance ideals

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