This paper is motivated by the maximization of k-th eigenvalue Laplace operator with Neumann boundary conditions among domains $${{\mathbb {R}}}^N$$ prescribed measure. We relax problem to class (possibly degenerate) densities in $$\mathbb {R}^N$$ mass and prove existence an optimal density. For $$k=1,2$$ , two problems are equivalent maximizers known be one equal balls, respectively. $$k \ge 3...

In this paper bounds for clusters of eigenvalues of non-selfadjoint matrices are investigated. We describe a method for the computation of rigorous error bounds for multiple or nearly multiple eigenvalues, and for a basis of the corresponding invariant subspaces. The input matrix may be real or complex, dense or sparse. The method is based on a quadratically convergent Newton-like method; it in...

Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map π : H → G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n “new” eigenvalues. We conjecture that every d-regular graph has a 2-lift such that all new eigenvalues are in the range [−2 √ d− 1, 2 √ d− 1] (If true, this is tight , e.g. by the Alon-Boppana bo...