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A conjecture by Brouwer and a conjecture by Grone and Merris state that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at most $e(G)+\\binom{k+1}{2}$ and $\\sum_{i=1}^{k}d_{i}^{*}$, respectively, where $(d_{i}^{*})_{i}$ is the conjugate of the degree sequence $(d_i)_{i}$. We generalize these conjectures to weighted graphs and symmetric matrices. 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