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

Abstract: Let be a distance-regular graph of diameter . Suppose that does not have any induced subgraph isomorphic to . In this case, the length of a shortest reduced circuit in is called the geometric girth of . Except for ordinary polygons, all known examples have a property that in general, and if . 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 with respect to a base vertex , which is called a subconstituent algebra or a Terwilliger algebra. The investigation of irreducible -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 -module is thin then is at most 8, and if , then and is a generalized octagon. In this paper, we prove the same result under an assumption that every irreducible -module of endpoint at most 3 is thin. [Copyright &y& Elsevier]