Bounds for sums of eigenvalues and applications

Let A be a matrix of order n × n with real spectrum λ1 ≥ λ2 ≥ ≥ λn. Let 1 ≤ k ≤ n − 2. If λn or λ1 is known, then we find an upper bound (respectively, lower bound) for the sum of the k-largest (respectively, k-smallest) remaining eigenvalues of A. Then, we obtain a majorization vector for (λ1, λ2,…, λn−1) when λn is known and a majorization vector for (λ2, λ3,…, λn) when λ1 is known. We apply these results to the eigenvalues of the Laplacian matrix of a graph and, in particular, a sufficient condition for a graph to be connected is given. Also, we derive an upper bound for the coefficient of ergodicity of a nonnegative matrix with real spectrum.