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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
##### Keywords:
random Laplacian matrix; Brouwer’s conjecture
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##### References:
 [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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