An inequality for regular near polygons

Let $G$ denote a near-polygon distance-regular graph with diameter $d\ge 3$, valency $k$ and intersection numbers $a_1>0$, $c_2>1$. Let ${\theta }_1$ denote the second largest eigenvalue for the adjacency matrix of $G$. We show ${\theta }_1$ is at most $(k-a_1-c_2)\mathrm{/}(c_2-1)$. We show the following are equivalent: (i) Equality is attained above; (ii) $G$ is $Q$-polynomial with respect to ${\theta }_1$; (iii) $G$ is a dual polar graph or a Hamming graph.