Fat Hoffman graphs with smallest eigenvalue at least $-1-\tau$

TL;DR

This paper classifies all fat Hoffman graphs with smallest eigenvalue at least -1- au, showing they can be constructed from a finite set of irreducible graphs, specifically 37 such graphs up to isomorphism.

Contribution

It provides a finite classification of fat Hoffman graphs with eigenvalue at least -1- au, identifying 37 irreducible graphs that generate all such graphs.

Findings

All such graphs are H-line graphs for a finite set H.

There are exactly 37 irreducible fat (-1- au)-Hoffman graphs.

The classification aids in understanding the spectral properties of these graphs.

Abstract

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.