On fat Hoffman graphs with smallest eigenvalue at least -3

TL;DR

This paper classifies fat Hoffman graphs with smallest eigenvalue at least -3 by analyzing their special graphs, showing they are represented by standard or irreducible root lattices, including Dynkin and extended Dynkin graphs.

Contribution

It provides a classification of fat Hoffman graphs with eigenvalue at least -3 based on their special graphs and lattice representations, linking graph theory with root lattice structures.

Findings

Special graph of an indecomposable fat Hoffman graph is represented by standard or irreducible root lattice.

If the special graph admits an integral representation, it is isomorphic to Dynkin or extended Dynkin graphs.

Classifies fat Hoffman graphs with eigenvalue ≥ -3 using lattice and graph-theoretic methods.

Abstract

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 A_n, D_n, or extended Dynkin graphs A_n or D_n.