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We derive a formula for the level spacing probability distribution in quantum graphs. We apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases. Moreover, we derive an…
We study the spectral statistics of quantum (metric) graphs whose vertices are equipped with preferred orientation vertex conditions. When comparing their spectral statistics to those predicted by suitable random matrix theory ensembles,…
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…
Motivated by the theory of quantum waveguides, we investigate the spectrum of the Laplacian, subject to Dirichlet boundary conditions, in a curved strip of constant width that is defined as a tubular neighbourhood of an infinite curve in a…
We review evolutionary models on quantum graphs expressed by linear and nonlinear partial differential equations. Existence and stability of the standing waves trapped on quantum graphs are studied by using methods of the variational…
We introduce a class of multiqubit quantum states which generalizes graph states. These states correspond to an underlying mathematical hypergraph, i.e. a graph where edges connecting more than two vertices are considered. We derive a…
We introduce a mapping between graphs and pure quantum bipartite states and show that the associated entanglement entropy conveys non-trivial information about the structure of the graph. Our primary goal is to investigate the family of…
The spectral properties of the Laplacian on a class of quantum graphs with random metric structure are studied. Namely, we consider quantum graphs spanned by the simple $\ZZ^d$-lattice with $\delta$-type boundary conditions at the vertices,…
We initiate a systematic study of quantum properties of finite graphs, namely, quantum asymmetry, quantum symmetry, and quantum isomorphism. We define the Schmidt alternative for a class of graphs, which reveals to be a useful tool for…
We review the theory of Cheeger constants for graphs and quantum graphs and their present and envisaged applications.
We review recent progress in understanding the physical meaning of quantum graph models through analysis of their vertex coupling approximations.
We present a systematic analysis of quantum Heisenberg-, XY- and interchange models on the complete graph. These models exhibit phase transitions accompanied by spontaneous symmetry breaking, which we study by calculating the generating…
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 spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs enjoys a rich group of symmetries, which enables us to obtain a decomposition of the corresponding Laplacian into the…
We examine transmission through a quantum graph vertex to which auxiliary edges with constant potentials are attached. We find a characterization of vertex couplings for which the transmission probability from a given "input" line to a…
A considerable success in phenomenological description of high-T$_{\rm c}$ superconductors has been achieved within the paradigm of Quantum Critical Point (QCP) - a parental state of a variety of exotic phases that is characterized by dense…
We investigate the parabolic Cauchy problem associated with quantum graphs including Lipschitz or polynomial type nonlinearities and additive Gaussian noise perturbed vertex conditions. The vertex conditions are the standard continuity and…
Quantum networks are important for quantum communication, enabling tasks such as quantum teleportation, quantum key distribution, quantum sensing, and quantum error correction, often utilizing graph states, a specific class of multipartite…
We consider the quantum site percolation model on graphs with an amenable group action. It consists of a random family of Hamiltonians. Basic spectral properties of these operators are derived: non-randomness of the spectrum and its…
We present a detailed study of the quantum site percolation problem on simple cubic lattices, thereby focussing on the statistics of the local density of states and the spatial structure of the single particle wavefunctions. Using the…