The Parameters of Bipartite Q-polynomial Distance-Regular GraphsJournal of Algebraic Combinatorics (2002)
AbstractLet Γ denote a bipartite distance-regular graph with diameter D ≥ 3 and valency k ≥ 3. Suppose θ0, θ1, ..., θ D is a Q-polynomial ordering of the eigenvalues of Γ. This sequence is known to satisfy the recurrence θ i − 1 − βθ i + θ i + 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.
Publication DateMay, 2002
Citation InformationJohn S Caughman. "The Parameters of Bipartite Q-polynomial Distance-Regular Graphs" Journal of Algebraic Combinatorics Vol. 15 Iss. 3 (2002)
Available at: http://works.bepress.com/john_caughman/16/