## Abstract

Let Γ denote a distance-regular graph with diameter d 3. By a parallelogram of length 3, we mean a 4-tuple xyzw consisting of vertices of Γ such that (x,y)= (z,w)=1, (x,z)=3, and (x,w)= (y,w)= (y,z)=2, where denotes the path-length distance function. Assume that Γ has intersection numbers a _{1}=0 and a _{2} 0. We prove that the following (i) and (ii) are equivalent. (i) Γ is Q-polynomial and contains no parallelograms of length 3; (ii) Γ has classical parameters (d,b,α,β) with b<-1. Furthermore, suppose that (i) and (ii) hold. We show that each of b(b+1)^{2}(b+2)/c _{2}, (b-2)(b-1)b(b+1)/(2+2b-c _{2}) is an integer and that c _{2} b(b+1). This upper bound for c _{2} is optimal, since the Hermitian forms graph Her_{2}(d) is a triangle-free distance-regular graph that satisfies c _{2}=b(b+1).

Original language | English |
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Pages (from-to) | 23-34 |

Number of pages | 12 |

Journal | Journal of Algebraic Combinatorics |

Volume | 27 |

Issue number | 1 |

DOIs | |

State | Published - 1 Feb 2008 |

## Keywords

- Classical parameters
- Distance-regular graph
- Q-polynomial