### Abstract

The critical group of a connected graph is a finite abelian group, whose order is the number of spanning trees in the graph, and which is closely related to the graph Laplacian. Its group structure has been determined for relatively few classes of graphs, e.g., complete graphs and complete bipartite graphs. For complete multipartite graphs K_{n1},...,n_{k}, we describe the critical group structure completely. For Cartesian products of complete graphs K_{n1}x⋯x K_{nk}, we generalize results of H. Bai on the k-dimensional cube, by bounding the number of invariant factors in the critical group, and describing completely its p-primary structure for all primes p that divide none of n_{1},..., n _{k}.

Original language | English (US) |
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Pages (from-to) | 231-250 |

Number of pages | 20 |

Journal | Journal of Graph Theory |

Volume | 44 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2003 |

### Keywords

- Abelian sandpile
- Cartesian product graph
- Chip-firing
- Complete multipartite graph
- Critical group
- Graph Laplacian
- Matrix-tree theorem
- Smith normal form

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## Cite this

*Journal of Graph Theory*,

*44*(3), 231-250. https://doi.org/10.1002/jgt.10139