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We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with $\delta$ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As…
We discuss Laplacian spectrum on a finite metric graph with vertex couplings violating the time-reversal invariance. For the class of star graphs we determine, under the condition of a fixed total edge length, the configurations for which…
We study the transmission of a quantum particle along a straight input--output line to which a graph $\Gamma$ is attached at a point. In the point of contact we impose a singularity represented by a certain properly chosen scale-invariant…
We investigate a periodic quantum graph in form of a square lattice with a general self-adjoint coupling at the vertices. We analyze the spectrum, in particular, its high-energy behaviour. Depending on the coupling type, bands and gaps have…
Complex periodic structures inherit spectral properties from the constituent parts of their unit cells, chiefly their spectral band gaps. Exploiting this intuitive principle, which is made precise in this work, means spectral features of…
A real band condition is shown to exist for one dimensional periodic complex non-hermitian potentials exhibiting PT-symmetry. We use an exactly solvable ultralocal periodic potential to obtain the band structure and discuss some spectral…
We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a…
We study spectral properties of a spinless quantum particle confined to an infinite planar layer with hard walls which interacts with a periodic lattice of point perturbations and a homogeneous magnetic field perpendicular to the layer. It…
In order to obtain perfect state transfer between two sites in a network of interacting qubits, their corresponding vertices in the underlying graph must satisfy a combinatorial property called strong cospectrality. Here we determine the…
We analytically study the effects of periodically alternating magnetic fields on the dynamics of a tight-binding ring. It is shown that an arbitrary quantum state can be frozen coherently at will by the very frequent square-wave field as…
This review paper summarizes the contents of the talk given by the author at the 8th International Congress of Chinese Mathematicians. Using examples of Schr\"odinger operators on metric graphs, it is shown that a nontrivial topology of the…
The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and…
We prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations, as predicted by random--matrix theory. The stability of these correlations with regard to non--universal corrections is analyzed in…
A general analytical approach to the statistical description of quantum graph spectra based on the exact periodic orbit expansions of quantum levels is discussed. The exact and approximate expressions obtained in \cite{Anima} for the…
We study theoretically the properties of buckled graphene-like materials, such as silicene and germanene, in a strong perpendicular magnetic field and a periodic potential. We analyze how the spin-orbit interaction and the perpendicular…
We present experimental and numerical results for the long-range fluctuation properties in the spectra of quantum graphs with chaotic classical dynamics and preserved time-reversal invariance. Such systems are generally believed to provide…
Boundary conditions in quantum graph vertices are generally given in terms of a unitary matrix $U$. Observing that if $U$ has at most two eigenvalues, then the scattering matrix $\mathcal{S}(k)$ of the vertex is a linear combination of the…
We consider a quantum ring of a certain radius R built from a sheet of the $\alpha$-$T_3$ lattice and solve for its spectral properties in presence of an external magnetic field. The energy spectrum consists of a conduction band, a valence…
We consider symmetric powers of a graph. In particular, we show that the spectra of the symmetric square of strongly regular graphs with the same parameters are equal. We also provide some bounds on the spectra of the symmetric squares of…
We study a set of scattering matrices of quantum graphs containing minimal number of passbands, i.e., maximal number of zero elements. The cases of even and odd vertex degree are considered. Using a solution of inverse scattering problem,…