On fat Hoffman graphs with smallest eigenvalue at least -3

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We investigate fat Hoffman graphs with smallest eigenvalue at least -3, using their special graphs. We show that the special graph S(H) of an indecomposable fat Hoffman graph H is represented by the standard lattice or an irreducible root lattice. Moreover, we show that if the special graph admits an integral representation, that is, the lattice spanned by it is not an exceptional root lattice, then the special graph S(H) is isomorphic to one of the Dynkin graphs Aₙ, Dₙ, or extended Dynkin graphs Aₙ or Dₙ.