Smith normal form and Laplacians

Let M denote the Laplacian matrix of a graph G. Associated with G is a finite group Φ ( G ) , obtained from the Smith normal form of M, and whose order is the number of spanning trees of G. We provide some general results on the relationship between the eigenvalues of M and the structure of Φ ( G )...

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Bibliographic Details
Published in:Journal of combinatorial theory. Series B 2008-11, Vol.98 (6), p.1271-1300
Main Author: Lorenzini, Dino
Format: Article
Language:English
Subjects:
Critical group
Eigenvalues
Graph
Group of components
Laplacian
Picard group
Sandpile group
Smith normal form
Strongly regular graph
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Summary:Let M denote the Laplacian matrix of a graph G. Associated with G is a finite group Φ ( G ) , obtained from the Smith normal form of M, and whose order is the number of spanning trees of G. We provide some general results on the relationship between the eigenvalues of M and the structure of Φ ( G ) , and address the question of how often the group Φ ( G ) is cyclic.
ISSN:0095-8956
1096-0902
DOI:10.1016/j.jctb.2008.02.002