Edge-signed graphs with smallest eigenvalue greater than −2

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 s...

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Bibliographic Details
Published in:Journal of combinatorial theory. Series B 2015-01, Vol.110, p.90-111
Main Authors: Greaves, Gary, Koolen, Jack, Munemasa, Akihiro, Sano, Yoshio, Taniguchi, Tetsuji
Format: Article
Language:English
Subjects:
Hoffman conjecture
Hoffman graphs
Line graphs
Signed graphs
Smallest eigenvalue
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Summary: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.
ISSN:0095-8956
1096-0902
DOI:10.1016/j.jctb.2014.07.006