Bounds for sums of eigenvalues and applications

Let A be a matrix of order n x n with real spectrum lambda(1) greater than or equal to lambda(2) greater than or equal to ... greater than or equal to lambda(n). Let 1 less than or equal to k less than or equal to n - 2. If l ambda(n) or lambda(1) is known, then we find an upper bound (respectively, lower bound) for the sum of the k-largest (respectively, k-smallest) remain ing eigenvalues of A. Then, we obtain a majorization vector for (lambda(1), lambda(2),...,lambda(n-1)) when lambda(n) is known and a majorization vecto r for (lambda(2), lambda(3), ... ,lambda(n)) when lambda(1) is known. We ap ply 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 give n. Also, we derive an upper bound for the coefficient of ergodicity of a no nnegative matrix with real spectrum. (C) 2000 Elsevier Science Ltd. All rig hts reserved.