The Parameters of Bipartite Q-polynomial Distance-Regular Graphs

  • Authors:

  • John S. Caughman, IV

  • Affiliations:

  • Department of Mathematical Sciences, Portland State University, P.O. Box 751, Portland, OR 97207-0751, USA. caughman@mth.pdx.edu

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2002

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Abstract

Let Γ denote a bipartite distance-regular graph with diameter iD ≥ 3 and valency ik ≥ 3. Suppose θ0, θ1, …, θiD is a iQ-polynomial ordering of the eigenvalues of Γ. This sequence is known to satisfy the recurrence θii − 1 − βθii + θii + 1 = 0 (0 i D), for some real scalar β. Let iq denote a complex scalar such that iq + iq−1 = β. Bannai and Ito have conjectured that iq is real if the diameter iD is sufficiently large.We settle this conjecture in the bipartite case by showing that iq is real if the diameter iD ≥ 4. Moreover, if iD = 3, then iq is not real if and only if θ1 is the second largest eigenvalue and the pair (μ, ik) 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.