Related papers: Spectral Estimates for Infinite Quantum Graphs
During the last years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wave function statistics. In the first part of this review we give a detailed introduction to…
For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$…
On a finite connected metric graph, we establish upper bounds for the eigenvalues of the Laplacian. These bounds depend on the length, the Betti number, and the number of pendant vertices. For trees, these estimates are sharp. We also…
Kirchhoff showed that the number of spanning trees of a graph is the spectral determinant of the combinatorial Laplacian divided by the number of vertices; we reframe this result in the quantum graph setting. We prove that the spectral…
We study the spectra of quantum graphs with the method of trace identities (sum rules), which are used to derive inequalities of Lieb-Thirring, Payne-P\'olya-Weinberger, and Yang types, among others. We show that the sharp constants of…
We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called riedrichs and Neumann extensions. We introduce a new…
We introduce a concept of isoperimetric dimension for magnetic graphs, that is, graphs where every edge is assigned a complex number of modulus one. In analogy with the classical case, we show that isoperimetric inequalities imply Sobolev…
The unitary evolution maps in closed chaotic quantum graphs are known to have universal spectral correlations, as predicted by random matrix theory. In chaotic graphs with absorption the quantum maps become non-unitary. We show that their…
Several new spectral properties of the normalized Laplacian defined for oriented hypergraphs are shown. The eigenvalue $1$ and the case of duplicate vertices are discussed; two Courant nodal domain theorems are established; new quantities…
In \cite{Elek} we proved that the limit of a weakly convergent sequence of finite graphs can be viewed as a graphing or a continuous field of infinite graphs. Thus one can associate a type $II_1$-von Neumann algebra to such graph sequences.…
We obtain several sharp spectral bounds, approximations, and exact values for the isoperimetric number and related edge-expansion parameters of graphs. Our results focus on graph powers and on families of graphs with rich algebraic or…
A method for estimating the spectral gap along with higher eigenvalues of nonequilateral quantum graphs has been introduced by Amini and Cohen-Steiner recently: it is based on a new transference principle between discrete and continuous…
We consider the inverse scattering on the quantum graph associated with the hexagonal lattice. Assuming that the potentials on the edges are compactly supported, we show that the S-matrix for all energies in any open set in the continuous…
We characterize the spectrum of the transition matrix for simple random walk on graphs consisting of a finite graph with a finite number of infinite Cayley trees attached. We show that there is a continuous spectrum identical to that for a…
We investigate the equivalence between spectral characteristics of the Laplace operator on a metric graph, and the associated unitary scattering operator. We prove that the statistics of level spacings, and moments of observations in the…
Quantum isometry groups of spectral triples associated with approximately finite-dimensional C*-algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated Bratteli diagrams. This is used to…
The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed.…
The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In…
Quantum graphs are an operator space generalization of classical graphs that have emerged in different branches of mathematics including operator theory, non-commutative topology and quantum information theory. In this paper, we obtain…
Quantum graphs have recently emerged as models of nonlinear optical, quantum confined systems with exquisite topological sensitivity and the potential for predicting structures with an intrinsic, off-resonance response approaching the…