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

Let ΓΓ be a distance-regular graph of diameter DD. Suppose that ΓΓ does not have any induced subgraph isomorphic to K2,1,1K2,1,1. In this case, the length of a shortest reduced circuit in ΓΓ is called the geometric girth gg(Γ) of ΓΓ. Except for ordinary polygons, all known examples have a property that gg(Γ)≤12 in general, and gg(Γ)≤8 if a1≠0a1≠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)T=T(x) with respect to a base vertex xx, which is called a subconstituent algebra or a Terwilliger algebra. The investigation of irreducible TT-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 TT-module is thin then gg(Γ) is at most 8, and if gg(Γ)=8, then a1=0a1=0 and ΓΓ is a generalized octagon. In this paper, we prove the same result under an assumption that every irreducible TT-module of endpoint at most 3 is thin.