On Q-polynomial distance-regular graphs with girth 6

Let Γ denote a Q-polynomial distance-regular graph with diameter D and valency k ≥ 3. By the result of H. Lewis, the girth of Γ  is at most 6. In this talk, we give a classification of graphs that attain this upper bound. We show that Γ  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).