Homotopy in Q - polynomial distance - regular graphs ( Heather

Let denote a Q-polynomial distance-regular graph with diameter d¿ 3. We show that if the valency is at least three, then the intersection number p 12 is at least two; consequently the girth is at most six. We then consider a condition on the dual eigenvalues of that must hold if is the quotient of an antipodal distance-regular graph of diameter D¿ 7; we call a pseudoquotient whenever this condition holds. For our main result, we show that if is not a pseudoquotient, then any cycle in can be ‘decomposed’ into cycles of length at most six. We present this result using homotopy. c © 2000 Elsevier Science B.V. All rights reserved.