  Let $G$ be a simple graph with the Laplacian matrix $L(G)$ and let $e(G)$ be
the number of edges of $G$. 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.
Moreover, among other results we show that under some assumptions, concave
upper bounds on convex functions of symmetric real matrices are equivalent to
concave upper bounds on convex functions of $(0,1)$ matrices.
