A classification of $Q$-polynomial distance-regular graphs with girth $6$

Let $\mathrm{\Gamma }$ denote a $Q$-polynomial distance-regular graph with diameter $D$ and valency $k\ge 3$. In [Homotopy in $Q$-polynomial distance-regular graphs, Discrete Math., {\bf 223} (2000), 189-206], H. Lewis showed that the girth of $\mathrm{\Gamma }$ is at most $6$. In this paper we classify graphs that attain this upper bound. We show that $\mathrm{\Gamma }$ has girth $6$ if and only if it is either isomorphic to the Odd graph on a set of cardinality $2D+1$, or to a generalized hexagon of order $(1,k-1)$.