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 graphsà n orD n .
There are many results on graphs with the smallest eigenvalue at least −2. As a next step, A. J. Hoffman proposed to study graphs with the smallest eigenvalue at least −1 − √ 2. In order to deal with such graphs, R. Woo and A. Neumaier introduced the concept of a Hoffman graph, and defined a new generalization of line graphs which depends on a family of Hoffman graphs. They proved a theorem analogous to Hoffman's, using a particular family consisting of four isomorphism classes.In this paper, we deal with a generalization based on a family H smaller than the one which they dealt with, yet including generalized line graphs in the sense of Hoffman. The main result is that the cover of an H -line graph with at least 8 vertices is unique.
Dedicated to Alan J. Hoffman on the occasion of his ninetieth birthday.Abstract. We give a structural classification of edge-signed graphs with smallest eigenvalue greater than −2. We prove a conjecture of Hoffman about the smallest eigenvalue of the line graph of a tree that was stated in the 1970s. Furthermore, we prove a more general result extending Hoffman's original statement to all edge-signed graphs with smallest eigenvalue greater than −2. Our results give a classification of the special graphs of fat Hoffman graphs with smallest eigenvalue greater than −3.
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