A NEW INEQUALITY FOR DISTANCE-REGULAR GRAPHS

Given a nontrivial primitive idempotent E of a distance-regular graph Gamma with diameter d greater than or equal to 3, we obtain an inequality involving the intersection numbers of Gamma for each integer i (3 less than or equal to i less than or equal to d). We show equality is attained for i = 3 if and only if equality is attained for all i (3 less than or equal to i less than or equal to d) if and only if Gamma is e-polynomial with respect to E. If the intersection numbers of Gamma are such that qc(i) - b(i) - q(qc(i-1) - b(i-1)) is independent of i (1 less than or equal to i less than or equal to d) for some q is an element of R\{0, - 1} (as is the case for many examples), our inequalities take an especially simple form.