Fat Hoffman graphs with smallest eigenvalue at least $-1-τ$

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

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