zbMATH — the first resource for mathematics

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.

05C80 Random graphs (graph-theoretic aspects)
05C22 Signed and weighted graphs
15A18 Eigenvalues, singular values, and eigenvectors
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.