The Brouwer spectral conjecture holds for all connected graphs whose vertex count is at least 4 times the square of the maximum degree; the ordinary-graph case also implies the loop/multigraph case.
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Remarks on the Brouwer Conjecture
The Brouwer spectral conjecture holds for all connected graphs whose vertex count is at least 4 times the square of the maximum degree; the ordinary-graph case also implies the loop/multigraph case.