  In this paper, we show that all fat Hoffman graphs with smallest eigenvalue
at least -1-\tau, where \tau is the golden ratio, can be described by a finite
set of fat (-1-\tau)-irreducible Hoffman graphs. In the terminology of Woo and
Neumaier, we mean that every fat Hoffman graph with smallest eigenvalue at
least -1-\tau is an H-line graph, where H is the set of isomorphism classes of
maximal fat (-1-\tau)-irreducible Hoffman graphs. It turns out that there are
37 fat (-1-\tau)-irreducible Hoffman graphs, up to isomorphism.
