Vertex Boundary Research Articles

Second-harmonic generation in branched optical waveguides: Metric graphs based approach

We consider the second-harmonic generation in branched optical waveguides, by modeling the process in terms of the three-wave interaction system on metric graphs. The vertex boundary conditions are derived from the conservation laws. Soliton solutions of the three-wave interaction system are obtained for the case of star- and tree-branched networks.

Transverse free vibration analysis of thin sectorial plates by the weak form quadrature element method

The weak form quadrature element method is applied to free vibration analysis of thin sectorial plates with arbitrary vertex angles and boundary conditions. To tackle the strong stress singularity around the vertex, analytical displacement descriptions are introduced into the inner sectorial subdomain, while the outer annular subdomain is modeled by a single weak form quadrature thin plate element. The continuity on the interface between the two subdomains is enforced afterwards. Eventually, a generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first six non-dimensional frequency parameters for various vertex angles and boundary conditions are obtained and compared with available results. Several typical free vibration modes are plotted. The accuracy, convergence rate, and computational cost of the present formulation are discussed at length.

Bogoliubov-de Gennes equation on graphs: A model for tree-branched Majorana wire network

We consider Bogoliubov-de Gennes equation on a metric tree graph. Formulation of the problem for arbitrary graph topology is provided. Self-adjoint vertex boundary conditions are derived. Exact solutions of the problem is obtained for quantum tree graph. A quantum graph based model for tree-branched Majorana wire network is proposed.

On transparent vertex boundary conditions for quantum graphs

On transparent vertex boundary conditions for quantum graphs

Fokker–Planck equation on metric graphs

We consider the Fokker–Planck equation on metric graphs. Vertex boundary conditions are imposed in the form of weight continuity and the probability current conservation. Exact solution of the Fokker–Planck equation on star, tree and loop graphs is obtained. Applications of the model to Brownian motion in networks and other problems are briefly discussed.

A note on Neumann problems on graphs

We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex boundary and provide analytic and probabilistic representations for Neumann solutions. A second result deals with Neumann problems on canonically compactifiable graphs with respect to the Royden boundary and provides conditions for unique solvability and analytic and probabilistic representations.

Concentration on the Boolean hypercube via pathwise stochastic analysis

We develop a new technique for proving concentration inequalities which relate the variance and influences of Boolean functions. Using this technique, we 1. Settle a conjecture of Talagrand (Combinatorica 17(2):275–285, 1997), proving that $$\begin{aligned} \int _{\left\{ -1,1\right\} ^{n}}\sqrt{h_{f}\left( x\right) }d\mu \left( x\right) \ge C\cdot \mathrm {Var}\left( f\right) \cdot \left( \log \left( \frac{1}{\sum \mathrm {Inf}_{i}^{2}\left( f\right) }\right) \right) ^{1/2}, \end{aligned}$$ where \(h_{f}\left( x\right) \) is the number of edges at x along which f changes its value, \(\mu \left( x\right) \) is the uniform measure on \(\left\{ -1,1\right\} ^{n}\), and \(\mathrm {Inf}_{i}\left( f\right) \) is the influence of the i-th coordinate. 2. Strengthen several classical inequalities concerning the influences of a Boolean function, showing that near-maximizers must have large vertex boundaries. An inequality due to Talagrand states that for a Boolean function f, \(\mathrm {Var}\left( f\right) \le C\sum _{i=1}^{n}\frac{\mathrm {Inf}_{i}\left( f\right) }{1+\log \left( 1/\mathrm {Inf}_{i}\left( f\right) \right) }\). We give a lower bound for the size of the vertex boundary of functions saturating this inequality. As a corollary, we show that for sets that satisfy the edge-isoperimetric inequality or the Kahn–Kalai–Linial inequality up to a constant, a constant proportion of the mass is in the inner vertex boundary. 3. Improve a quantitative relation between influences and noise stability given by Keller and Kindler. Our proofs rely on techniques based on stochastic calculus, and bypass the use of hypercontractivity common to previous proofs.

Nonlocal nonlinear Schrödinger equationon metric graphs: A model for generation and transport of parity-time-symmetric nonlocal solitons in networks.

We consider the parity-time (PT)-symmetric, nonlocal, nonlinear Schrödinger equationon metric graphs. Vertex boundary conditions are derived from the conservation laws. Soliton solutions are obtained for the simplest graph topologies, such as star and tree graphs. The integrability of the problem is shown by proving the existence of an infinite number of conservation laws. A model for soliton generation in such PT-symmetric optical fibers and their networks governed by the nonlocal nonlinear Schrödinger equation is proposed. Exact formulas for the number of generated solitons are derived for the cases when the problem is integrable. Numerical solutions are obtained for the case when integrability is broken.

Planar Lattice Subsets with Minimal Vertex Boundary

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice $X$. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We show that the neighborhood of a minimal set is minimal. We characterize uniquely minimal sets of $X$: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of $X$: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate the graph $G$ of minimal sets whose vertices are congruence classes of minimal sets of $X$ and whose edges connect vertices which can be represented by minimal sets that differ by exactly one vertex. We prove that G has exactly one infinite component, has infinitely many isolated vertices and has bounded components of arbitrarily large size. Finally, we show that all minimal sets, except one, are connected.

A new approach to finding the extra connectivity of graphs

The connectivity of a graph is the minimum number of nodes, whose removal will cause the graph disconnected. It is one of the basic and important properties of the graph, and can be used as a metric of the graph’s robustness. Since the connectivity of a graph is upper-bounded by its minimal node-degree, which can be small, the notion of conditional connectivity has been proposed in the past to more realistically reflect a graph’s robustness. The extra connectivity is such a conditional connectivity. In this paper, we introduce a new approach to finding the extra connectivity of the hypercube, a well-known regular graph model for networks of parallel computers. We will show that the results on the isoperimetric inequalities for hypercubes can be used to obtain their extra connectivity. That is, using isoperimetric inequalities, for an n-dimensional hypercube Qn with n≥5, the h-extra connectivity is equal to its minimum (h+1)-vertex boundary number for 1≤h≤n−4 and n+1≤h≤2n−4. For n−3≤h≤n, the h-extra connectivity of hypercube equals its minimum (n−2)-vertex boundary number. The paper’s result for the first time establishes a relationship between the isoperimetric inequalities and the extra connectivity of graphs.

Separable Reversible Data Hiding for Encrypted Three-Dimensional Models Based on Spatial Subdivision and Space Encoding

Reversible data hiding for encrypted media not only preserves the privacy of the media content but also can convey additional information during message transmission. Some studies advocate separability, that is, the message can be correctly extracted irrespective of whether the encrypted media has been decrypted. However, current research has focused on encrypted images. Urgent research is required on encrypted three-dimensional (3D) models. This paper proposes a separable reversible data hiding method based on spatial subdivision and space encoding for encrypted 3D models. A bounding volume is first constructed using the vertices with boundary values in the processing model. Each vertex coordinate value is then converted into a ratio (between 0 and 1) of the distance between the vertex and minimum boundary point to the side length of the bounding volume. The owner of the 3D model then uses a secret key to encrypt all ratios except those of the boundary vertices to obtain an encrypted 3D model of the same size as the original model. The spatial subdivision technique and a subdivision threshold are subsequently used to divide the bounding volume into a series of blocks and simultaneously control the vertex distortion. The secret message is embedded in the encrypted vertex by using the space encoding method with an embedding threshold. Experimental results indicate that the proposed algorithm enables high privacy, performs separable reversible data hiding, and has low computational complexity, high embedding capacity, and controllable distortion.

Stability for vertex isoperimetry in the cube

We prove a stability version of Harper's cube vertex isoperimetric inequality, showing that subsets of the cube with vertex boundary close to the minimum possible are close to (generalised) Hamming balls. Furthermore, we obtain a local stability result for ball-like sets that gives a sharp estimate for the vertex boundary in terms of the distance from a ball, and so our stability result is essentially tight (modulo a non-monotonicity phenomenon). We also give similar results for the Kruskal–Katona Theorem and applications to new stability versions of some other results in Extremal Combinatorics.

Transparent vertex boundary conditions for quantum graphs: Simplified approach

Transparent vertex boundary conditions for quantum graphs: Simplified approach

Transparent quantum graphs

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 Schrödinger equation on metric graphs. This allows to derive simple constraints, which make usual Kirchhoff-type boundary conditions at the vertex equivalent to the transparent ones. The approach is applied to quantum star and tree graphs. However, extension to more complicated graph topologies is rather straightforward.

Bogoliubov de Gennes equation on metric graphs

We consider Bogoliubov de Gennes equation on metric graphs. The vertex boundary conditions providing self-adjoint realization of the Bogoliubov de Gennes operator on a metric star graph are derived. Secular equation providing quantization of the energy and the vertex transmission matrix are also obtained. Application of the model for Majorana wire networks is discussed.

The stationary sine-Gordon equation on metric graphs: Exact analytical solutions for simple topologies

We consider the stationary sine-Gordon equation on metric graphs with simple topologies. Exact analytical solutions are obtained for different vertex boundary conditions. It is shown that the method can be extended for tree and other simple graph topologies. Applications of the obtained results to branched planar Josephson junctions and Josephson junctions with tricrystal boundaries are discussed.

Dynamic analysis in three layered conical shells utilising numerical methods

In this work, dynamic composite analysis truncated sandwich conical shells (STCS) with different boundary conditions under low velocity impact was investigated analytically, according higher order sandwich panel theory. Impact occurred at the top face sheet and the contact force history is predicted using two solution models calculated applying principal of Hamilton by considering the displacement continuity conditions between the layers. For obtaining the displacement and contact force histories, the differential quadrature method (DQM) is utilised. In this investigation, the effects of layer lamina number, cone semi vertex angle, boundary conditions and impact velocity on the impact response are studied.

Dynamic analysis in three layered conical shells utilising numerical methods

In this work, dynamic composite analysis truncated sandwich conical shells (STCS) with different boundary conditions under low velocity impact was investigated analytically, according higher order sandwich panel theory. Impact occurred at the top face sheet and the contact force history is predicted using two solution models calculated applying principal of Hamilton by considering the displacement continuity conditions between the layers. For obtaining the displacement and contact force histories, the differential quadrature method (DQM) is utilised. In this investigation, the effects of layer lamina number, cone semi vertex angle, boundary conditions and impact velocity on the impact response are studied.

The Growth Constant of Odd Cutsets in High Dimensions

A cutset is a non-empty finite subset of ℤd which is both connected and co-connected. A cutset is odd if its vertex boundary lies in the odd bipartition class of ℤd. Peled [18] suggested that the number of odd cutsets which contain the origin and have n boundary edges may be of order eΘ(n/d) as d → ∞, much smaller than the number of general cutsets, which was shown by Lebowitz and Mazel [15] to be of order dΘ(n/d). In this paper, we verify this by showing that the number of such odd cutsets is (2+o(1))n/2d.

Quantum search on the two-dimensional lattice using the staggered model with Hamiltonians

Quantum search on the two-dimensional lattice with one marked vertex and cyclic boundary conditions is an important problem in the context of quantum algorithms with an interesting unfolding. It avails to test the ability of quantum walk models to provide efficient algorithms from the theoretical side and means to implement quantum walks in laboratories from the practical side. In this paper, we rigorously prove that the recent-proposed staggered quantum walk model provides an efficient quantum search on the two-dimensional lattice, if the reflection operators associated with the graph tessellations are used as Hamiltonians, which is an important theoretical result for validating the staggered model with Hamiltonians. Numerical results show that on the two-dimensional lattice staggered models without Hamiltonians are not as efficient as the one described in this paper and are, in fact, as slow as classical random-walk-based algorithms.