Small clique number graphs with three trivial critical ideals
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 generaliz...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2018-03-01
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| Series: | Special Matrices |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/spma-2018-0011 |
| Summary: | 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. |
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| ISSN: | 2300-7451 |