# Perkel graph

In mathematics, the **Perkel graph**, named after Manley Perkel, is a 6-regular graph with 57 vertices and 171 edges. It is the unique distance-regular graph with intersection array (6, 5, 2; 1, 1, 3).[1] The Perkel graph is also distance-transitive.

Perkel graph | |
---|---|

Perkel graphs with 19-fold symmetry | |

Vertices | 57 |

Edges | 171 |

Radius | 3 |

Diameter | 3 |

Girth | 5 |

Automorphisms | 3420 |

Chromatic number | 3 |

Properties | Regular, distance-transitive |

Table of graphs and parameters |

It is also the skeleton of an abstract regular polytope, the 57-cell.

## References

- Coolsaet, K. and Degraer, J. "A Computer Assisted Proof of the Uniqueness of the Perkel Graph." Designs, Codes and Crypt. 34, 155–171, 2005.

- Brouwer, A. E.
*Perkel Graph.*. - Brouwer, A. E.; Cohen, A. M.; and Neumaier, A.
*The Perkel Graph for L(2,19).*13.3 in Distance Regular Graphs. New York: Springer-Verlag, pp. 401–403, 1989. - Perkel, M.
*Bounding the Valency of Polygonal Graphs with Odd Girth.*Can. J. Math. 31, 1307-1321, 1979. - Perkel, M.
*Characterization of in Terms of Its Geometry.*Geom. Dedicata 9, 291-298, 1980.

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