Related papers: Nodal domains on quantum graphs

We present the results of experimental study of nodal domains of wave functions (electric field distributions) lying in the regime of Shnirelman ergodicity in the chaotic half-circular microwave rough billiard. Nodal domains are regions…

We treat the stationary nonlinear Schroodinger equation on two-dimensional branched domains, so-called fat graphs. The shrinking limit when the domain becomes one-dimensional metric graph is studied by using analytical estimate of the…

A Random Geometric Graph (RGG) ensemble is defined by the disordered distribution of its node locations. We investigate how this randomness drives sample-to-sample fluctuations in the dynamical properties of these graphs. We study the…

In this thesis, we study Laplacian eigenfunctions on metric graphs, also known as quantum graphs. We restrict the discussion to standard quantum graphs. These are finite connected metric graphs with functions that satisfy Neumann vertex…

Let $G$ be a random $d$-regular graph. We prove that for every constant $\alpha > 0$, with high probability every eigenvector of the adjacency matrix of $G$ with eigenvalue less than $-2\sqrt{d-2}-\alpha$ has $\Omega(n/$polylog$(n))$ nodal…

Let N(f) be a number of nodal domains of a random Gaussian spherical harmonic f of degree n. We prove that as n grows to infinity, the mean of N(f)/n^2 tends to a positive constant, and that N(f)/n^2 exponentially concentrates around that…

We establish various nodal domain theorems for $p$-Laplacians on signed graphs, which unify most of the existing results on nodal domains of graph $p$-Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we…

Power grids, road maps, and river streams are examples of infrastructural networks which are highly vulnerable to external perturbations. An abrupt local change of load (voltage, traffic density, or water level) might propagate in a…

Random geometric graphs consist of randomly distributed nodes (points), with pairs of nodes within a given mutual distance linked. In the usual model the distribution of nodes is uniform on a square, and in the limit of infinitely many…

Let $G$ be a simple, connected graph on $n$ vertices, and further assume that $G$ has disjoint cycles. Let $h$ be a real symmetric matrix supported on $G$ (for example, a discrete Schr\"odinger operator). The eigenvalues of $h$ are ordered…

We prove sharp upper bounds for eigenvalues of Schr\"odinger operators on quantum graphs with $\delta$-coupling (also known as Robin) conditions at all vertices. The bounds depend on the geometry of the graph, on the potential, and the…

We consider a variant of so called power-law random graph. A sequence of expected degrees corresponds to a power-law degree distribution with finite mean and infinite variance. In previous works the asymptotic picture with number of nodes…

We consider families of finite quantum graphs of increasing size and we are interested in how eigenfunctions are distributed over the graph. As a measure for the distribution of an eigenfunction on a graph we introduce the entropy, it has…

Let $H(\Om_0)=-\Delta+V$ be a Schr\"odinger operator on a bounded domain $\Om_0\subset \mathbb R^d$ with Dirichlet boundary conditions. Suppose that the $\Om_\ell$ ($\ell \in \{1,...,k\}$) are some pairwise disjoint subsets of $\Om_0$ and…

Exchangeable random graphs serve as an important probabilistic framework for the statistical analysis of network data. In this work we develop an alternative parameterization for a large class of exchangeable random graphs, where the nodes…

We consider a periodic system of domains coupled by small windows. In such domain we study the band spectrum of a Schroedinger operator subject to Neumann condition. We show that near each isolated eigenvalue of the similar operator but in…

The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the…

We study the nodal set of eigenfunctions of the Laplace operator on the right angled isosceles triangle. A local analysis of the nodal pattern provides an algorithm for computing the number of nodal domains for any eigenfunction. In…

We show that the variance of the number of connected components of the zero set of the two-dimensional Gaussian ensemble of random spherical harmonics of degree n grows as a positive power of n. The proof uses no special properties of…

We give several algebraic bounds for percolation on directed and undirected graphs: proliferation of strongly-connected clusters, proliferation of in- and out-clusters, and the transition associated with the number of giant components.