Small clique number graphs with three trivial critical ideals-Reference-Cited by-同舟云学术

Small clique number graphs with three trivial critical ideals

Published:2018-03-01 Issue:1 Volume:6 Page:122-154
ISSN:2300-7451
Container-title:Special Matrices
language:en
Short-container-title:

Author:

Alfaro Carlos A.¹,Valencia Carlos E.²

Affiliation:

1. Banco de México, Calzada Legaria 691, módulo IV, Col. Irrigación, 11500 Mexico City , D.F., Mexico

2. Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14–740, 07000 Mexico City , Mexico

Abstract

Abstract The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. Previously, they have been used in the understanding and characterizing of the graphs with critical group with few invariant factors equal to one. However, critical ideals generalize the critical group, Smith group and the characteristic polynomials of the adjacency and Laplacian matrices of a graph. In this article we provide a set of minimal forbidden graphs for the set of graphs with at most three trivial critical ideals. Then we use these forbidden graphs to characterize the graphs with at most three trivial critical ideals and clique number equal to 2 and 3.

Publisher

Walter de Gruyter GmbH

Subject

Geometry and Topology,Algebra and Number Theory

Link

https://www.degruyter.com/document/doi/10.1515/spma-2018-0011/pdf

Reference26 articles.

1. [1] C.A. Alfaro, J. C.-H. Lin. Critical ideals, minimum rank and zero forcing number. Preprint arXiv:1710.03386.

2. [2] C.A. Alfaro, C.E. Valencia, A. Vázquez-Ávila. Digraphs with at most one trivial critical ideal. Linear Multilinear Algebra, 2017. DOI: http://dx.doi.org/10.1080/03081087.2017.1383349

3. [3] C.A. Alfaro and C.E. Valencia, On the sandpile group of the cone of a graph. Linear Algebra Appl. 436 (2012) 5 1154-1176. MR2890910

4. [4] C.A. Alfaro and C.E. Valencia, Graphs with two trivial critical ideals. Discrete Appl. Math. 167 (2014) 33-44.

5. [5] C.A. Alfaro, H.H. Corrales and C.E. Valencia, Critical ideals of signed graphs with twin vertices. Adv. in Appl.Math. 86 (2017) 99-131.

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