An inequality for regular near polygons

Let Gamma denote a near polygon distance-regular graph with diameter d greater than or equal to 3, valency k and intersection numbers a(1) > 0, c(2) > 1. Let theta(1) denote the second largest eigenvalue of Gamma. We show theta(1) less than or equal to k - a(1) - c(2)/ c(2) - 1. We show the following (i)-(iii) are equivalent. (i) Equality is attained above; (ii) Gamma is Q-polynomial with respect to theta(1); (iii) Gamma is a dual polar graph or a Hamming graph. (C) 2004 Elsevier Ltd. All rights reserved.