Graphs with few trivial characteristic ideals

We give a characterization of the graphs with at most three trivial characteristic ideals. This implies the complete characterization of the regular graphs whose critical groups have at most three invariant factors equal to 1 and the characterization of the graphs whose Smith groups have at most 3 i...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 615; pp. 155 - 174
Main Authors Alfaro, Carlos A., Barrus, Michael D., Sinkovic, John, Villagrán, Ralihe R.
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 15.04.2021
American Elsevier Company, Inc
Subjects
Characteristic ideal
Critical group
Forbidden induced subgraph
Graphs
Invariants
Linear algebra
Smith group
Forbidden induced subgraph
05C50
13C40
13F20
Critical group
Smith group
05C25
Characteristic ideal
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Summary:We give a characterization of the graphs with at most three trivial characteristic ideals. This implies the complete characterization of the regular graphs whose critical groups have at most three invariant factors equal to 1 and the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1. We also give an alternative and simpler way to obtain the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1, and a list of minimal forbidden graphs for the family of graphs with Smith group having at most 4 invariant factors equal to 1.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2020.12.024