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

34222

Integral Tree N=31 (Brouwer's #17)

Kevin Ryde

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

Acyclic | Yes | Index | 3 |

Algebraic Connectivity | 0.108 | Laplacian Largest Eigenvalue | 7.826 |

Average Degree | 1.935 | Longest Induced Cycle | undefined |

Bipartite | Yes | Longest Induced Path | 6 |

Chromatic Index | 6 | Matching Number | 10 |

Chromatic Number | 2 | Maximum Degree | 6 |

Circumference | undefined | Minimum Degree | 1 |

Claw-Free | No | Minimum Dominating Set | 10 |

Clique Number | 2 | Number of Components | 1 |

Connected | Yes | Number of Edges | 30 |

Density | 0.065 | Number of Triangles | 0 |

Diameter | 6 | Number of Vertices | 31 |

Edge Connectivity | 1 | Planar | Yes |

Eulerian | No | Radius | 3 |

Genus | 0 | Regular | No |

Girth | undefined | Second Largest Eigenvalue | 2 |

Hamiltonian | No | Smallest Eigenvalue | -3 |

Independence Number | 21 | Vertex Connectivity | 1 |

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

Posted by Kevin Ryde at Jan 5, 2020 1:50 AM.

This tree is integral, meaning all roots of the characteristic polynomial of its adjacency matrix are integers. It was found by Brouwer and is one of 3 integral trees of n=31 vertices.

A. E. Brouwer, "Small Integral Trees", The Electronic Journal of Combinatorics, volume 15, 2008. Figure 1 picture, and table 1 depths and spectrum (tree #17, the new n=31). http://www.win.tue.nl/~aeb/graphs/integral_trees.html, http://www.win.tue.nl/~aeb/preprints/small_itrees.pdf

Also appears in A. E. Brouwer and W. H. Haemers, "Spectra of Graphs", Springer, 2011, section 5.7 exercise 1, page 88. http://www.win.tue.nl/~aeb/2WF02/spectra.pdf

A. E. Brouwer, "Small Integral Trees", The Electronic Journal of Combinatorics, volume 15, 2008. Figure 1 picture, and table 1 depths and spectrum (tree #17, the new n=31). http://www.win.tue.nl/~aeb/graphs/integral_trees.html, http://www.win.tue.nl/~aeb/preprints/small_itrees.pdf

Also appears in A. E. Brouwer and W. H. Haemers, "Spectra of Graphs", Springer, 2011, section 5.7 exercise 1, page 88. http://www.win.tue.nl/~aeb/2WF02/spectra.pdf

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