Sum of the largest eigenvalues of symmetric matrices and graphs

D. Gernert conjectured that the sum of two largest eigenvalues of the adjacency matrix of a simple graph is at most the number of vertices of the graph. This can be proved, in particular, for all regular graphs. Gernert’s conjecture was recently disproved by Nikiforov, who also provided a nontrivial upper bound for the sum of two largest eigenvalues. We will study extensions of these results to general n×n symmetric matrices with entries from [0, 1] and try to improve Nikiforov’s theorem. We will also study other recent results on the extreme behavior of the sum the k largest eigenvalues of symmetric matrices and particularly, adjacency matrices of graphs.