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 |
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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