Homotopy in Q-polynomial distance-regular graphs

Abstract

Let Γ 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 3 is at least two; consequently the girth is at most six. We then consider a condition on the dual eigenvalues of Γ that must hold if Γ is the quotient of an antipodal distance-regular graph of diameter D⩾7; we call Γ a pseudoquotient whenever this condition holds. For our main result, we show that if Γ is not a pseudoquotient, then any cycle in Γ can be ‘decomposed’ into cycles of length at most six. We present this result using homotopy.