## Abstract

If a nonsymmetric P-polynomial association scheme, or equivalently, a distance-regular digraph, has diameter d and girth g, then d = g or d = g - 1, by Damerell's theorem. The dual of this theorem was proved by Leonard. In this paper, we prove that the diameter of a nonsymmetric P- and Q-polynomial association scheme is one less than its girth and its cogirth. We also give a structure theorem for a nonsymmetric Q-polynomial association scheme whose diameter is equal to its cogirth. We use self-duality and unimodality to show that the eigenvalues of a nontrivial nonsymmetric P- and Q-polynomial association scheme are quadratic over the rationals. The fact that the adjacency algebra becomes a C-algebra gives a necessary condition for the existence of a nonsymmetric P- and Q-polynomial association scheme. As an application, it is shown that the only nontrivial nonsymmetric P- and Q-polynomial association scheme with girth 5 is the directed 5 cycle.

Original language | English |
---|---|

Pages (from-to) | 314-328 |

Number of pages | 15 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 51 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1991 Mar |

Externally published | Yes |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics