The Parameters of Bipartite Q-polynomial Distance-Regular Graphs
نویسنده
چکیده
Let denote a bipartite distance-regular graph with diameter D ≥ 3 and valency k ≥ 3. Suppose θ0, θ1, . . . , θD is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence θi−1 − βθi + θi+1 = 0 (0 < i < D), for some real scalar β. Let q denote a complex scalar such that q + q−1 = β. Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large. We settle this conjecture in the bipartite case by showing that q is real if the diameter D ≥ 4. Moreover, if D = 3, then q is not real if and only if θ1 is the second largest eigenvalue and the pair (μ, k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.
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