On fat Hoffman graphs with smallest eigenvalue at least -3

Hye Jin Jang, Jack Koolen, Akihiro Munemasa, Tetsuji Taniguchi

Abstract


We investigate fat Hoffman graphs with smallest eigenvalue at least −3, using their special graphs. We show that the special graph S(Ho) of an indecomposable fat Hoffman graph Ho 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 − (Ho) is isomorphic to one of the Dynkin graphs An, Dn, or extended Dynkin graphs Ãn or D̃n.


Keywords


Hoffman graph, line graph, graph eigenvalue, special graph, root system

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DOI: https://doi.org/10.26493/1855-3974.262.a9d

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications