Abstract If ${\mathbf v} \in {\mathbb R}^{V(X)}$ is an eigenvector for eigenvalue $\lambda $ of a graph X and $\alpha automorphism , then ({\mathbf v})$ also . Thus, it rather exceptional vertex-transitive to have multiplicity one. We study cubic graphs with nontrivial simple eigenvalue, discover remarkable connections arc-transitivity, regular maps, number theory.

The problem of partitioning a graph such that the number of edges incident to vertices in diierent partitions is minimized, arises in many contexts. Some examples include its recursive application for minimizing ll-in in matrix factorizations and load-balancing for parallel algorithms. Spectral graph partitioning algorithms partition a graph using the eigenvector associated with the second smal...

This note gives converses to the well-known result: for any vector e u such that sin (u; e u) = O(), we have e u Ae u e u e u = + O(2) where is an eigenvalue and u is the corresponding eigenvector of a Her-mitian matrix A, and \ " denotes complex conjugate transpose. It shows that if e u Ae u=e u e u is close to A's largest eigenvalue, then e u is close to the corresponding eigenvector with an ...

Two different Lanczos-type methods for the linear response eigenvalue problem are analyzed. The first one is a natural extension of the classical Lanczos method for the symmetric eigenvalue problem while the second one was recently proposed by Tsiper specially for the linear response eigenvalue problem. Our analysis leads to bounds on errors for eigenvalue and eigenvector approximations by the ...

2.1 Computing Eigenvectors from the Characteristic Polynomial The characteristic polynomial can be used to develop an algorithm for computing the eigenvectors and eigenvalues. This algorithm, which is known as Power Iteration, takes advantage of the property that vectors which are transformed by the matrix A will be scaled in the direction of the largest eigenvector. Initially, the vector is ch...

For an eigenvalue λ0 of a Hermitian matrix A, the formula Thompson and McEnteggert gives explicit expression adjugate λ0I−A, Adj(λ0I−A), in terms eigenvectors A for all its eigenvalues. In this paper Thompson-McEnteggert's is generalized to include any with entries arbitrary field. addition, nonsingular elementary divisors Adj(A) provided those A. Finally, generalization eigenvalue-eigenvector ...

We give a necessary and sufficient condition for a graph to be bipartite in terms of an eigenvector corresponding to the largest eigenvalue of the adjacency matrix of the graph.

The almost Mathieu operator is the discrete Schrödinger operator Hα,β,θ on `2(Z) defined via (Hα,β,θf)(k) = f(k+1)+f(k−1)+β cos(2παk+θ)f(k). We derive explicit estimates for the eigenvalues at the edge of the spectrum of the finite-dimensional almost Mathieu operator H α,β,θ. We furthermore show that the (properly rescaled) m-th Hermite function φm is an approximate eigenvector of H α,β,θ, and ...