Graphs whose critical groups have larger rank

Yao Ping Hou; Wai Chee Shiu; Wai Hong Chan

doi:10.1007/s10114-011-9358-6

Graphs whose critical groups have larger rank

Yao Ping Hou, Wai Chee Shiu, Wai Hong Chan

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

7 Citations (Scopus)

Abstract

The critical group C(G) of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r(G) be the minimum number of generators (i.e., the rank) of the group C(G) and β(G) be the number of independent cycles of G. In this paper, some forbidden induced subgraphs are given for r(G) = n - 3 and all graphs with r(G) = β(G) = n - 3 are characterized.

Original language	English
Pages (from-to)	1663-1670
Number of pages	8
Journal	Acta Mathematica Sinica, English Series
Volume	27
Issue number	9
DOIs	https://doi.org/10.1007/s10114-011-9358-6
Publication status	Published - Sept 2011

Scopus Subject Areas

General Mathematics
Applied Mathematics

User-Defined Keywords

Critical group of a graph
Laplacian matrix
Smith normal form

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10.1007/s10114-011-9358-6

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abstract = "The critical group C(G) of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r(G) be the minimum number of generators (i.e., the rank) of the group C(G) and β(G) be the number of independent cycles of G. In this paper, some forbidden induced subgraphs are given for r(G) = n - 3 and all graphs with r(G) = β(G) = n - 3 are characterized.",

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AU - Hou, Yao Ping

AU - Shiu, Wai Chee

AU - Chan, Wai Hong

N1 - Funding Information: Received June 25, 2009, accepted January 18, 2010 Supported by FRG, Hong Kong Baptist University; the first author is supported by National Natural Science Foundation of China (Grant No. 10671061)

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N2 - The critical group C(G) of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r(G) be the minimum number of generators (i.e., the rank) of the group C(G) and β(G) be the number of independent cycles of G. In this paper, some forbidden induced subgraphs are given for r(G) = n - 3 and all graphs with r(G) = β(G) = n - 3 are characterized.

AB - The critical group C(G) of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r(G) be the minimum number of generators (i.e., the rank) of the group C(G) and β(G) be the number of independent cycles of G. In this paper, some forbidden induced subgraphs are given for r(G) = n - 3 and all graphs with r(G) = β(G) = n - 3 are characterized.

KW - Critical group of a graph

KW - Laplacian matrix

KW - Smith normal form

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JF - Acta Mathematica Sinica, English Series

IS - 9

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Graphs whose critical groups have larger rank

Abstract

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