The gap between 1/2 and the sum of squared Perron vector entries on an independent set can be exponentially small in graphs with arbitrarily large chromatic number, disproving two conjectures.
Haemers, Hoffman’s ratio bound, Linear Algebra Appl
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An exponentially small gap of the Perron vector on independent sets
The gap between 1/2 and the sum of squared Perron vector entries on an independent set can be exponentially small in graphs with arbitrarily large chromatic number, disproving two conjectures.