The energy of a graph was introduced by Gutman in 1978 as the sum of the absolute values of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. These are Cayley graphs on cyclic groups (i.e. there adjacency matrix is circulant) each of whose eigenvalues is an integer. Given an arbitrary prime power p, we determine all integral circu...

Glossary Dilation matrix A 2× 2 matrix M is called a dilation matrix if all the entries of M are integers and all the eigenvalues of M are greater than one in modulus. Isotropic dilation matrix A dilation matrix M is said to be isotropic if M is similar to a diagonal matrix and all its eigenvalues have the same modulus. Wavelet system A wavelet system is a collection of square integrable functi...

We show that the random matrix theory with non-integer " symmetry parameter " β describes the statistics of transport parameters of strongly disordered two dimensional systems. The application of the random matrix theory (RMT) [1] to electronic transport in weakly disordered systems [2] enables us to understand the main features of the transport. The small number of parameters which enters RMT ...

Let π1, π2 be cuspidal automorphic representations of PGL2(R) of conductor 1 and Hecke eigenvalues λπ1,2 (n), and let h > 0 be an integer. For any smooth compactly supported weight functions W1,2 : R → C and any Y > 0 a spectral decomposition of the shifted convolution sum

We perform high-order sensitivity analysis of eigenvalues of linear systems depending on parameters. Attention is focused on double not-semi-simimple and semi-simple eigenvalues, undergoing perturbations, either of regular or singular type. The use of integer (Taylor) or fractional (Puiseux) series expansions is discussed, and the analysis carried out on the characteristic polynomial. It is sho...

Sets which simultaneously tile Rn by applying powers of an invertible matrix and translations by a lattice are studied. Diagonal matrices A for which there exist sets that tile by powers of A and by integer translations are characterized. A sufficient condition and a necessary condition on the dilations and translations for the existence of such sets are also given. These conditions depend in a...

We study the behavior of certain eigenvalues for magnetic Aharonov-Bohm operators with half-integer circulation and Dirichlet boundary conditions in a planar domain. We analyze the leading term in the Taylor expansion of the eigenvalue function as the pole moves in the interior of the domain, proving that it is a harmonic homogeneous polynomial and determining its exact coefficients.

We solve for the spectrum and eigenfunctions of Dirac operator on the sphere. The eigenvalues are nonzero whole numbers. The eigenfunctions are two-component spinors which may be classified by representations of the SU (2) group with half-integer angular momenta. They are special linear combinations of conventional spherical spinors.

A classic result is that the number of distinct eigenvalues of a graph of diameter d is at least d+1. More generally, let G be a graph with n vertices, and let S(G) denote the set of all symmetric n by n matrices, A = [aij ], with the property that for off-diagonal entries, aij is nonzero if and only if G has an edge from i to j. The traditional proof of the classic result extends to A ∈ S(G). ...

It is well-known that the Artin-Mazur dynamical zeta function of a hyperbolic or quasi-hyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. We give an elementary proof of this fact that extends to the case of general toral endomorphisms without change. The result is a closed formula that can be calculated ...