On fat Hoffman graphs with smallest eigenvalue at least -3

Details On fat Hoffman graphs with smallest eigenvalue at least -3 On fat Hoffman graphs with smallest eigenvalue at least -3

2011-10-31

arxiv.org/abs/1110.6821v1

Mathematics

Combinatorics

21 pages

Scientific paper

Hoffman graphs are a limiting object of graphs with respect to the smallest eigenvalue. To understand graphs with smallest eigenvalue -3, 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 tilde{A}_n or tilde{D}_n.

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