A new inequality for bipartite distance-regular

Let Gamma denote a bipartite distance-regular graph with diameter D greater than or equal to 4 and valency k greater than or equal to 3. Let theta denote an eigenvalue of Gamma other than k and -k and consider the associated cosine sequence, sigma(0,)sigma(1),..,sigma(D.). We show (sigma(1) - sigma(i+1))(sigma(1) - sigma(i-1))greater than or equal to(sigma(2) - sigma(i))(sigma(0) - sigma(i)) for 1 less than or equal to i less than or equal to D - 1. We show the following are equivalent: (i) equality is attained above for i = 3, (ii) equality is attained above for 1 less than or equal to i less than or equal to D - 1, (iii) there exists a real scalar beta such that sigma(i- 1) - betasigma(i) + sigma(i+1) is independent of i for 1 less than or equal to i less than or equal to D - 1. We say theta is three-term recurrent (or TTR) whenever (i)-(iii) are satisfied. We discuss the connection between TTR eigenvalues and the Q-polynomial property. When an eigenvalue is TTR, we find formulae for the intersection numbers and eigenvalues of Gamma in terms of at most two free parameters, classifying Gamma if beta = +/-2. Among the eigenvalues of Gamma in their natural order, we consider which can be TTR. We show Gamma can have at most three distinct TTR eigenvalues. We show Gamma has three distinct TTR eigenvalues if and only if Gamma is 2-homogeneous in the sense of Curtin and Nomura. We show Gamma has exactly two distinct TTR eigenvalues if and only if Gamma is antipodal with diameter 5, but not 2-homogeneous. (C) 2003 Elsevier Inc. All rights reserved.