Does anyone know if there is a reference for the spectrum of the cocktail party graph? How can I find the eigenvalues of this famous graph?
2026-02-23 00:27:34.1771806454
Spectrum of cocktail party graph?
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For any $k\in\omega$, the spectrum of the hyperoctahedral graph (aka cocktail party) graph on $n=2k$ vertices is known to be
(Here, ${}^\times$ means repeated juxtaposition, not multiplication.)
A reference is [J. L. Gross, J. Yellen, Handbook of Graph Theory, CRC Press, 2004, ISBN 9780203490204, page 559].
One should also note that since each cocktail party graph is (vertex-degree-) regular, there is a useful connection of its spectrum to the spectrum of its complement, and this complement is a perfect matching, whose spectrum is particularly easy to understand.