Spectra of bipartite P- and Q-polynomial association schemes

Let Y = (X, {R-i}(0 less than or equal to i less than or equal to D)) denote a symmetric association scheme with D greater than or equal to 3, and assume Y is not an ordinary cycle. Suppose Y is bipartite P-polynomial with respect to the given ordering A(0), A(1),..., A(D) of the associate matrices, and Q-polynomial with respect to the ordering E-0, E-1,..., E-D of the primitive idempotents. Then the eigenvalues and dual eigenvalues satisfy exactly one of (i)-(iv). (i) theta(0) > theta(1) > theta(2) > theta(3) > ... > theta(D-3) > theta(D-2) > theta(D-1) > theta(D), theta(i)* = theta(i) (0 less than or equal to i less than or equal to D). (ii) D is even, and theta(0) > theta(D-1) > theta(2) > theta(D-3) >... > theta(3) > theta(D-2) > theta(1) > theta(D), theta(i)* = theta(i) (0 less than or equal to i less than or equal to D). (iii) theta(0)* > theta(0), and theta(0) > theta(1) > theta(2) > theta(3) >...> theta(D-3) > theta(D-2) > theta(D-1) > theta(D), theta(0)* > theta(1)* > theta(2)* > theta(3)* > ... > theta*(D-3) > theta*(D-2) > theta*(D-1) > theta*(D). (iv) theta(0)* > theta(0), D is odd, and theta(0) > theta(D-1) > theta(2) > theta(D-3) >...> theta(3) > theta(D-2) > theta(1) > theta(D), theta(0)* > theta(D)* > theta(2)* > theta(D-2)* >...> theta(D-3)* > theta(3)* > theta(D-1)* > theta(1)*.