Explicit construction of families of linear expanders and superconcentrators is relevant to theoretical computer science in several ways. There is essentially only one known explicit construction. Here we show a correspondence between the eigenvalues of the adjacency matrix of a graph and its expansion properties, and combine it with results on Group Representations to obtain many new examples ...

2 Eigenvalues of graphs 5 2.1 Matrices associated with graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The largest eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Adjacency matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.3...

We denote by G = (V,E) a d-regular graph on n = |V | nodes with no multiple edges and no self-loops. We denote by A its adjacency matrix. Thus, A is a symmetric n×n-matrix in which each element either is zero or one and each row and column contains exactly d ones and n− d zeros. We denote by d = μ1 ≥ . . . ≥ μn the eigenvalues of A, and we denote by d = λ1 ≥ . . . ≥ λn the corresponding absolut...

In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey. ∗This article appeared in Combinatorial and Graph-Theoretical Problems in Linear Algebra, R. A. Brualdi, S. Friedland, V. Klee, Eds., IMA Volumes in Mathema...

Upper bounds for eigenvalues of a solution to continuous time coupled algebraic Riccati equation (CCARE) and discrete time coupled algebraic Riccati equation (DCARE) are developed as special cases of bounds for the uni...ed coupled algebraic Riccati equation (UCARE). They include bounds of the maximal eigenvalues, the sums of the eigenvalues and the trace.

We consider the eigenvalues of the biharmonic operator subject to several homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show that simple eigenvalues and elementary symmetric functions of multiple eigenvalues are real analytic, and provide Hadamard-type formulas for the corresponding shape derivatives. After recalling the known results in shape optimization, we prove ...

2 Let H = [ M K K∗ N ] be a Hermitian matrix. It is known that the eigenvalues of M ⊕N are 3 majorized by the eigenvalues of H . If, in addition, H is positive semidefinite and the block K 4 is Hermitian, then the following reverse majorization inequality holds for the eigenvalues: 5

It is shown that on every finite network with at least one circuit there exist second order differential operators having an infinite number of nonreal eigenvalues. The presence of nonreal eigenvalues implies that these operators cannot be selfadjoint with respect to any metric. These eigenvalues reveal also the existence of oscillatory solutions for the corresponding time–dependent partial dif...

We revisit Hoffman relation involving chromatic number χ and eigenvalues. We construct some graphs and weighted graphs such that the largest and smallest eigenvalues λ dan μ satisfy λ = (1 − χ)μ. We study in particular the eigenvalues of the integer simplex T 2 m, a 3-chromatic graph on ( m+2 2 ) vertices.