Graphs whose critical groups have larger rank

Yao Ping HOU, Wai Chee SHIU, Wai Hong CHAN

Research output: Contribution to journalArticle

4 Citations (Scopus)


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. Copyright © 2011 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.
Original languageEnglish
Pages (from-to)1663-1670
JournalActa Mathematica Sinica, English Series
Issue number9
Publication statusPublished - Sep 2011


Critical Group
Systems science
Graph in graph theory
Forbidden Induced Subgraph
Laplacian Matrix
Spanning tree


Hou, Y. P., Shiu, W. C., & Chan, W. H. (2011). Graphs whose critical groups have larger rank. Acta Mathematica Sinica, English Series, 27(9), 1663-1670. doi: 10.1007/s10114-011-9358-6


  • Critical group of a graph
  • Laplacian matrix
  • Smith normal form