## Abstract

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph. • The first deals with connected graphs containing no cut-edge. Here the number of independent cycles in the graph, which is known to bound the number of generators for the critical group of the graph, is shown also to bound the number of generators for the critical group of its line graph. • The second gives, for each prime p, a constraint on the p-primary structure of the critical group, based on the largest power of p dividing all sums of degrees of two adjacent vertices. • The third deals with connected graphs whose line graph is regular. Here known results relating the number of spanning trees of the graph and of its line graph are sharpened to exact sequences which relate their critical groups. The first two results interact extremely well with the third. For example, they imply that in a regular nonbipartite graph, the critical group of the graph and that of its line graph determine each other uniquely in a simple fashion.

Original language | English (US) |
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Pages (from-to) | 449-488 |

Number of pages | 40 |

Journal | Annals of Combinatorics |

Volume | 16 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2012 |

### Bibliographical note

Funding Information:∗ Work of all authors supported by NSF grants DMS-0245379 and DMS-0601010, and completed partly during REU programs at the University of Minnesota during the summers of 2003, 2004, and 2008.

## Keywords

- critical group
- line graph
- regular graph