Brouwer's conjecture

In the mathematical field of spectral graph theory, Brouwer's conjecture is a conjecture by Andries Brouwer on upper bounds for the intermediate sums of the eigenvalues of the Laplacian of a graph in term of its number of edges.^[1]

The conjecture states that if G is a simple undirected graph and L(G) its Laplacian matrix, then its eigenvalues λ_n(L(G)) ≤ λ_n−1(L(G)) ≤ ... ≤ λ₁(L(G)) satisfy $\sum _{i=1}^{t}\lambda _{i}(L(G))\leq m(G)+\left({\begin{array}{c}t+1\\2\end{array}}\right),\quad t=1,\ldots ,n$ where m(G) is the number of edges of G.

^ Brouwer, Andries E.; Haemers, Willem H. (2012). Spectra of Graphs. Universitext. New York, NY: Springer New York. doi:10.1007/978-1-4614-1939-6. ISBN 978-1-4614-1938-9.

[:0-1] Brouwer, Andries E.; Haemers, Willem H. (2012). Spectra of Graphs. Universitext. New York, NY: Springer New York. doi:10.1007/978-1-4614-1939-6. ISBN 978-1-4614-1938-9.

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