The spectra of the local graphs of the twisted Grassmann graphs

In this paper we will determine the spectra of the local graphs of the twisted Grassmann graphs and look at its consequences for the associated Terwilliger algebras. In particular, we show that the Terwilliger algebra for the twisted Grassmann graphs does depend on the base point. More specifically, we show that for the twisted Grassmann graphs there are two vertices x and y such that for the Terwilliger algebra with base point x all the irreducible modules are thin and for the Terwilliger algebra with base point y there exist non-thin irreducible modules.