Parallelogram-Free Distance-Regular Graphs

LetΓ=(X, R) denote a distance-regular graph with distance function ∂ and diameterd⩾4. By a parallelogram of lengthi(2⩽i⩽d), we mean a 4-tuplexyzuof vertices inXsuch that ∂(x, y)=∂(z, u)=1, ∂(x, u)=i, and ∂(x, z)=∂(y, z)=∂(y, u)=i−1. We prove the following theorem.Theorem.LetGamma;denote a distance-regular graph with diameterd⩾4, and intersection numbersa1=0,a2≠0. SupposeΓisQ-polynomial and contains no parallelograms of length 3 and no parallelograms of length 4. ThenΓ:has classical parameters (d, b, α,β) withb<−1. By including results in [3], [9], we have the following corollary.Corollary. LetGammadenote a distance-regular graph with theQ-polynomial property. Suppose the diameterd⩾4. Then the following (i)–(ii) are equivalent. (i)Γcontains no parallelograms of any length. (ii) One of the following (iia)–(iic) holds. (iia)Γis bipartite. (iib)Γis a generalized odd graph. (iic)Γhas classical parameters (d, b, α, β) and eitherb<−1 orΓis a Hamming graph or a dual polar graph.