## Adjacency matrix[Too large to display] |
## Adjacency list[Too large to display] |

32256

Bulgarian Solitaire N=8

Kevin Ryde

Invariant | Value | Invariant | Value |
---|---|---|---|

Acyclic | No | Index | 2.399 |

Algebraic Connectivity | 0 | Laplacian Largest Eigenvalue | 5.137 |

Average Degree | 1.909 | Longest Induced Cycle | 4 |

Bipartite | Yes | Longest Induced Path | 9 |

Chromatic Index | 3 | Matching Number | 10 |

Chromatic Number | 2 | Maximum Degree | 3 |

Circumference | 4 | Minimum Degree | 1 |

Claw-Free | No | Minimum Dominating Set | 9 |

Clique Number | 2 | Number of Components | 2 |

Connected | No | Number of Edges | 21 |

Density | 0.091 | Number of Triangles | 0 |

Diameter | infinity | Number of Vertices | 22 |

Edge Connectivity | 0 | Planar | Yes |

Eulerian | No | Radius | undefined |

Genus | undefined | Regular | No |

Girth | 4 | Second Largest Eigenvalue | 2.034 |

Hamiltonian | No | Smallest Eigenvalue | -2.399 |

Independence Number | 12 | Vertex Connectivity | 0 |

A table row rendered like this
indicates that the graph is marked as being *interesting* for that invariant.

Posted by Kevin Ryde at Nov 29, 2018 7:52 AM.

Each vertex is a partition of N=8, ie. a set of integers which sum to 8. There are 22 such. Each edge is from a partition to its successor under the Bulgarian solitaire rule: subtract 1 from each element, sum those 1s as a new term, discard any zeros.

N=8 is the first where the resulting graph is not connected. It has 2 components (OEIS A037306). One ends in a 4-cycle. The other ends in a 2-cycle. That 2-cycle is a single edge in the simple graph here.

N=8 is the first where the resulting graph is not connected. It has 2 components (OEIS A037306). One ends in a 4-cycle. The other ends in a 2-cycle. That 2-cycle is a single edge in the simple graph here.

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