An inequality for regular near polygons

Let Γ denote a near polygon distance-regular graph with diameter d≥3, valency k and intersection numbers a1>0, c2>1. Let θ1 denote the second largest eigenvalue of Γ. We showθ1≤k−a1−c2c2−1.We show the following (i)–(iii) are equivalent. (i) Equality is attained above; (ii) Γ is Q-polynomial with respect to θ1; (iii) Γ is a dual polar graph or a Hamming graph.