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Brouwer’s conjecture holds asymptotically almost surely. (English) Zbl 1437.05217
Summary: We show that for a sequence of random graphs Brouwer’s conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs with possible negative weights as well. For graphs with a fixed number of vertices, the result implies that there are constants \(C > 0\) and \(n_0\) such that if \(n \geq n_0\) then among all \(2^{\binom{ n}{2}}\) graphs with \(n\) vertices, at least \((1 - \exp(- C n)) 2^{\binom{ n}{2}}\) graphs satisfy Brouwer’s conjecture.

MSC:
05C80 Random graphs (graph-theoretic aspects)
05C22 Signed and weighted graphs
15A18 Eigenvalues, singular values, and eigenvectors
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[1] Brouwer, Andries E.; Haemers, Willem H., Spectra of Graphs (2012), Springer-Verlag: Springer-Verlag New York · Zbl 1231.05001
[2] Ding, Xue; Jiang, Tiefeng, Spectral distributions of adjacency and Laplacian matrices of random graphs, Ann. Appl. Probab., 20, 6, 2086-2117 (2010) · Zbl 1231.05236
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