Related papers: Distinguishing co-spectral quantum graphs by scatt…

The scattering amplitude in simple quantum graphs is a well-known process which may be highly complex. In this work, motivated by the Shannon entropy, we propose a methodology that associates to a graph a scattering entropy, which we call…

Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of…

Observational data usually comes with a multimodal nature, which means that it can be naturally represented by a multi-layer graph whose layers share the same set of vertices (users) with different edges (pairwise relationships). In this…

In the broad range of studies related to quantum graphs, quantum graph spectra appear as a topic of special interest. They are important in the context of diffusion type problems posed on metric graphs. Theoretical findings suggest that…

In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of…

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 propose an approach to quantize discrete networks (graphs with discrete edges). We introduce a new exact solution of discrete Schrodinger equation that is used to write the solution for quantum graphs. Formulation of the problem and…

We prove that for a pair of cospectral graphs G and H, there exist their non trivial lifts G0 and H0 which are cospectral. More over for a pair of cospectral graphs on 6 vertices, we find some cospectral lifts of them.

In this paper, we use graph theory to solve wave scattering problems in the discrete dipole approximation. As a key result of this work, in the presence of active scatterers, we present a systematic method to find arbitrary large-order zero…

We construct explicit families of graphs whose eigenvalues are asymptotically distributed according to Wigner's semicircle law; in other words, that are spectrally indistinguishable from random graphs. However, in other respects they are…

We generalize the scattering approach to quantum graphs to quantum graphs with with piecewise constant potentials and multiple excitation modes. The free single-mode case is well-known and leads to the trace formulas of Roth, Kottos and…

Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices…

The explicit solution to the spectral problem of quantum graphs is used to obtain the exact distributions of several spectral statistics, such as the oscillations of the quantum momentum eigenvalues around the average, $\delta…

A finite dimensional operator that commutes with some symmetry group admits quotient operators, which are determined by the choice of associated representation. Taking the quotient isolates the part of the spectrum supporting the chosen…

The scattering process on multiloop infinite p+1-valent graphs (generalized trees) is studied. These graphs are discrete spaces being quotients of the uniform tree over free acting discrete subgroups of the projective group $PGL(2, {\bf…

We introduce the concept of regular quantum graphs and construct connected quantum graphs with discrete symmetries. The method is based on a decomposition of the quantum propagator in terms of permutation matrices which control the way…

We present enumeration results on the number of connected graphs up to 10 vertices for which there is at least one other graph with the same spectrum (a cospectral mate), or at least one other graph with the same Smith normal form…

We investigate the quantum networks that their nodes are considered as quantum harmonic oscillators. The entanglement of the ground state can be used to quantify the amount of information one part of a network shares with the other part of…

Representing a graph as a vector is a challenging task; ideally, the representation should be easily computable and conducive to efficient comparisons among graphs, tailored to the particular data and analytical task at hand. Unfortunately,…

We consider quantum graphs with transparent branching points. To design such networks, the concept of transparent boundary conditions is applied to the derivation of the vertex boundary conditions for the linear Schrodinger equation on…