Fat Hoffman graphs with smallest eigenvalue at least −1 − τ

Abstract

In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least − 1 − τ, where τ is 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 − τ is 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.