The geometric girth of a distance-regular graph having certain thin irreducible modules for the Terwilliger algebra

Let Γ be a distance-regular graph of diameter D . Suppose that Γ does not have any induced subgraph isomorphic to K 2 , 1 , 1 . In this case, the length of a shortest reduced circuit in Γ is called the geometric girth gg ( Γ ) of Γ . Except for ordinary polygons, all known examples have a property that gg ( Γ ) ≤ 12 in general, and gg ( Γ ) ≤ 8 if a 1 ≠ 0 . Is there an absolute constant bound on the geometric girth of a distance-regular graph with valency at least three? This is one of the main problems in the field of distance-regular graphs. P. Terwilliger defined an algebra T = T ( x ) with respect to a base vertex x , which is called a subconstituent algebra or a Terwilliger algebra. The investigation of irreducible T -modules and their thin property proved to be a very important tool to study structures of distance-regular graphs. B. Collins proved that if every irreducible T -module is thin then gg ( Γ ) is at most 8, and if gg ( Γ ) = 8 , then a 1 = 0 and Γ is a generalized octagon. In this paper, we prove the same result under an assumption that every irreducible T -module of endpoint at most 3 is thin.