On the sum of k largest eigenvalues of graphs and symmetric matrices

Let k be a positive integer and let G be a graph of order n ⩾ k . It is proved that the sum of k largest eigenvalues of G is at most 1 2 ( k + 1 ) n . This bound is shown to be best possible in the sense that for every k there exist graphs whose sum is 1 2 ( k + 1 2 ) n − o ( k − 2 / 5 ) n . A generalization to arbitrary symmetric matrices is given.