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The eigenvalues of oppositeness graphs in buildings of spherical type. (English) Zbl 1232.05122
Brualdi, Richard A. (ed.) et al., Combinatorics and graphs. Selected papers based on the presentations at the 20th anniversary conference of IPM on combinatorics, Tehran, Iran, May 15–21, 2009. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4865-4/pbk). Contemporary Mathematics 531, 1-10 (2010).
Summary: We consider the graph \(\Gamma\) obtained by taking as vertices the flags in a finite building of spherical type defined over \(\mathbb{F}_q\), where two flags are adjacent when they are opposite. We show that the squares of the eigenvalues of \(\Gamma\) are powers of \(q\).
For the entire collection see [Zbl 1202.05003].

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05Exx Algebraic combinatorics
20E42 Groups with a \(BN\)-pair; buildings
20G40 Linear algebraic groups over finite fields
51N30 Geometry of classical groups