Periodic Graphs Research Articles

Trapped modes in thin and infinite ladder like domains. Part 2: Asymptotic analysis and numerical application

We are interested in a 2D propagation medium obtained from a localized perturbation of a reference homogeneous periodic medium. This reference medium is a thick graph , namely a thin structure (the thinness being characterized by a small parameter e > 0) whose limit (when e tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. In the first part of this work, we proved that such a geometrical perturbation is able to produce localized eigenmodes (the propagation model under consideration is the scalar Helmholtz equation with Neumann boundary conditions). This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We used a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough. The objective of the present work is to complement the previous one by constructing and justifying a high order asymptotic expansion of these eigenvalues (with respect to the small parameter e) using the method of matched asymptotic expansions. In particular, the obtained expansion can be used to compute a numerical approximation of the eigenvalues and of their associated eigenvectors. An algorithm to compute each term of the asymptotic expansion is proposed. Numerical experiments validate the theoretical results.

From clusters to crystals: scale chemistry of intermetallics

We review different approaches to description, classification, and understanding of intermetallic structures of any complexity. All these approaches are based on the representation of the intermetallic structure as a periodic graph (net). A special attention is paid to the nanocluster model, which enables one to separate structural units and to describe their assemblage in intermetallics with strict computer algorithms. We demonstrate the abilities of the model with analysis of 12,315 intermetallic structures from the Inorganic Crystal Structure Database.

Dispersion relations of periodic quantum graphs associated with Archimedean tilings (II)

We continue the work of a previous paper (Luo et al 2019 J. Phys. A. Math. Theor. 52 165201) to derive the dispersion relations of the periodic quantum graphs associated with the remaining 5 of the 11 Archimedean tilings, namely the truncated hexagonal tiling (3,122), rhombi-trihexagonal tiling , snub square tiling (32,4,3,6), snub trihexagonal tiling (34,6), and truncated trihexagonal tiling . The computation is done with the help of the symbolic software Mathematica. With these explicit dispersion relations, we perform more analysis on the spectra.

Book Embeddings of Infinite Sequences of Extended Periodic Regular Graphs

A graph is extended periodic if it can be made up by gluing identical pieces together with a graph which is isomorphic to or subgraph of that piece, in a circular fashion. We consider an infinite sequence Hn of extended periodic 4-regular graphs and an infinite sequence Mn of extended periodic 5-regular graphs. A book is a collection of half-planes called pages, all having a common boundary called the spine. A book embedding of a graph G is to put the vertices of G on the spine and each edge of G is contained in one page. The minimum pages needed to embed a graph G into a book is called pagenumber of G. We determine the pagenumbers of Hn and Mn to be three.

Construction of breather solutions for nonlinear Klein-Gordon equations on periodic metric graphs

The purpose of this paper is to construct small-amplitude breather solutions for a nonlinear Klein-Gordon equation posed on a periodic metric graph via spatial dynamics and center manifold reduction. The major difficulty occurs from the irregularity of the solutions. The persistence of the approximately constructed pulse solutions under higher order perturbations is obtained by symmetry and reversibility arguments.

Reducible Fermi surfaces for non-symmetric bilayer quantum-graph operators

This work constructs a class of non-symmetric periodic Schrödinger operators on metric graphs (quantum graphs) whose Fermi, or Floquet, surface is reducible. The Floquet surface at an energy level is an algebraic set that describes all complex wave vectors admissible by the periodic operator at the given energy. The graphs in this study are obtained by coupling two identical copies of a periodic quantum graph by edges to form a bilayer graph. Reducibility of the Floquet surface for all energies ensues when the coupling edges have potentials belonging to the same asymmetry class. The notion of asymmetry class is defined in this article through the introduction of an entire spectral A-function a(\lambda) associated with a potential – two potentials belong to the same asymmetry class if their A-functions are identical. Symmetric potentials correspond to a(\lambda) \equiv 0 . If the potentials of the connecting edges belong to different asymmetry classes, then typically the Floquet surface is not reducible. An exception occurs when two copies of certain bipartite graphs are coupled; the Floquet surface in this case is always reducible. This includes AA-stacked bilayer graphene.

Mass-constrained ground states of the stationary NLSE on periodic metric graphs

We investigate the existence of ground states with fixed mass for the nonlinear Schrödinger equation with a pure power nonlinearity on periodic metric graphs. Within a variational framework, both the $$L^2$$ -subcritical and critical regimes are studied. In the former case, we establish the existence of global minimizers of the NLS energy for every mass and every periodic graph. In the critical regime, a complete topological characterization is derived, providing conditions which allow or prevent ground states of a certain mass from existing. Besides, a rigorous notion of periodic graph is introduced and discussed.

Spectral estimates for Schrödinger operators on periodic discrete graphs

We consider normalized Laplacians and their perturbations by periodic potentials (Schr\"odinger operators) on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of non-degenerate bands) and a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graphs and show that they become identities for some class of graphs. We determine two-sided estimates on the length of the first spectral band and on the effective mass at the bottom of the spectrum of the Laplace and Schr\"odinger operators. In particular, these estimates yield that the first spectral band of Schr\"odinger operators is non-degenerate.

Invariants for Laplacians on periodic graphs

We consider a Laplacian on periodic discrete graphs. Its spectrum consists of a finite number of bands. In a class of periodic 1-forms, i.e., functions defined on edges of the periodic graph, we introduce a subclass of minimal forms with a minimal number $\cI$ of edges in their supports on the period. We obtain a specific decomposition of the Laplacian into a direct integral in terms of minimal forms, where fiber Laplacians (matrices) have the minimal number $2\cI$ of coefficients depending on the quasimomentum and show that the number $\cI$ is an invariant of the periodic graph. Using this decomposition, we estimate the position of each band, the Lebesgue measure of the Laplacian spectrum and the effective masses at the bottom of the spectrum in terms of the invariant $\cI$ and the minimal forms. In addition, we consider an inverse problem: we determine necessary and sufficient conditions for matrices depending on the quasimomentum on a finite graph to be fiber Laplacians. Moreover, similar results for Schr\"odinger operators with periodic potentials are obtained.

Elliptic differential operators on infinite graphs with general conditions on vertices

ABSTRACTLet Γ be an unbounded metric-oriented graph embedded in , , be countable sets of edges , and vertices of Γ. The graph Γ is equipped with a differential equation(1) with piece-wise smooth coefficients , and general conditions at the vertices (2) where is the number of edges incident to v, are complex matrices, are limit values at the vertex of the derivatives taken along the edges according to their orientation. We associate with equation (1) and the vertex conditions (2) an operator acting from the Sobolev space to the space where . We study the smoothness, and exponential behavior at infinity of solutions of equation , and for periodic graphs we obtain the necessary and sufficient conditions of the Fredholmness of and a description of the essential spectrum of the realization of in We give applications of these results to the Schrödinger operators on periodic graphs with general conditions at the vertices.

Electromagnetic Schrödinger Operators on Periodic Graphs with General Conditions at Vertices

The main aim of the paper is the study of the Fredholm property and essential spectra of electromagnetic Schrodinger operators $$\mathcal{H}$$ on graphs periodic with respect to a group $$\mathbb{G}$$ isomorphic to ℤk. We consider the Schrodinger operators with nonperiodic electric and magnetic potentials and with general nonperiodic conditions on the vertices.We associate with $$\mathcal{H}$$ a family Lim( $$\mathcal{H}$$ ) of limit operators $$\mathcal{H}$$ h generated by sequences h: $$\mathbb{G}$$ ∍hm → ∞. The main results of the paper are: (i) $$\mathcal{H}$$ is a Fredholm operator if and only if all limit operators $$\mathcal{H}$$ h of $$\mathcal{H}$$ are invertible, (ii) 1 $$s{p_{ess}}H\, = \,\bigcup\limits_{{H^h} \in Lim\left( H \right)} {sp{H^h}} $$ where spess $$\mathcal{H}$$ is the essential spectrum of $$\mathcal{H}$$ . Formula (1) is applied to the study of the essential spectrum of Schrodinger operators whose potentials are perturbations of periodic magnetic and electric potentials by slowly oscillating terms.

Dispersion relations of periodic quantum graphs associated with Archimedean tilings (I)

There are 11 kinds of Archimedean tilings for the plane in total. Applying the Floquet–Bloch theory, we derive the dispersion relations of the periodic quantum graphs associated with a number of Archimedean tilings, namely the triangular tiling (36), the elongated triangular tiling , the trihexagonal tiling and the truncated square tiling (4,82). The derivation makes use of characteristic functions, with the help of the symbolic software Mathematica. The resulting dispersion relations are surprisingly simple and symmetric. They show that in each case the spectrum is composed of point spectrum and an absolutely continuous spectrum. We further analyzed the structure of absolutely continuous spectra. Our work is motivated by the studies on the periodic quantum graphs associated with hexagonal tiling in Kuchment and Post (2007 Commun. Math. Phys. 275 805–26) and Korotyaev and Lobanov (2007 Ann. Henri Poincaré 8 1151–76).

Gaps in the Spectrum of a Cuboidal Periodic Lattice Graph

We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with $\delta$ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As the main result, we find a connection between the arrangement of the gaps and the coefficients in a continued fraction associated with the ratio of edge lengths of the lattice. This knowledge enables a straightforward construction of a periodic quantum graph with any required number of spectral gaps and---to some degree---to control their positions; i.e., to partially solve the inverse spectral problem.

Determinantal spanning forests on planar graphs

We generalize the uniform spanning tree to construct a family of determinantal measures on essential spanning forests on periodic planar graphs in which every component tree is bi-infinite. Like the uniform spanning tree, these measures arise naturally from the Laplacian on the graph. More generally, these results hold for the “massive” Laplacian determinant which counts rooted spanning forests with weight $M$ per finite component. These measures typically have a form of conformal invariance, unlike the usual rooted spanning tree measure. We show that the spectral curve for these models is always a simple Harnack curve; this fact controls the decay of edge-edge correlations in these models. We construct a limit shape theory in these settings, where the limit shapes are defined by measured foliations of fixed isotopy type.

Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs

We investigate the existence of ground states for the focusing nonlinear Schroedinger equation on a prototypical doubly periodic metric graph. When the nonlinearity power is below 4, ground states exist for every value of the mass, while, for every nonlinearity power between 4 (included) and 6 (excluded), a mark of $L^2$-criticality arises, as ground states exist if and only if the mass exceeds a threshold value that depends on the power. This phenomenon can be interpreted as a continuous transition from a two-dimensional regime, for which the only critical power is 4, to a one-dimensional behavior, in which criticality corresponds to the power 6. We show that such a dimensional crossover is rooted in the coexistence of one-dimensional and two-dimensional Sobolev inequalities, leading to a new family of Gagliardo-Nirenberg inequalities that account for this continuum of critical exponents.

Coordination sequences and layer-by-layer growth of periodic structures

Abstract A new approach to the problem of coordination sequences of periodic structures is proposed. It is based on the concept of layer-by-layer growth and on the study of geodesics in periodic graphs. We represent coordination numbers as sums of so called sector coordination numbers arising from the growth polygon of the graph. In each sector we obtain a canonical form of the geodesic chains and reduce the calculation of the sector coordination numbers to solution of the linear Diophantine equations. The approach is illustrated by the example of the 2-homogeneous kra graph. We obtain three alternative descriptions of the coordination sequences: explicit formulas, generating functions and recurrent relations.

Diffusive stability for periodic metric graphs

AbstractWe consider a nonlinear diffusion equation on an infinite periodic metric graph. We prove that the terms which are irrelevant w.r.t. linear diffusion on the real line are irrelevant w.r.t. linear diffusion on the periodic metric graph, too. The proof is based on L1‐ estimates combined with Bloch wave analysis for periodic metric graphs.

Learning Urban Community Structures

Learning urban community structures refers to the efforts of quantifying, summarizing, and representing an urban community’s (i) static structures, e.g., Point-Of-Interests (POIs) buildings and corresponding geographic allocations, and (ii) dynamic structures, e.g., human mobility patterns among POIs. By learning the community structures, we can better quantitatively represent urban communities and understand their evolutions in the development of cities. This can help us boost commercial activities, enhance public security, foster social interactions, and, ultimately, yield livable, sustainable, and viable environments. However, due to the complex nature of urban systems, it is traditionally challenging to learn the structures of urban communities. To address this problem, in this article, we propose a collective embedding framework to learn the community structure from multiple periodic spatial-temporal graphs of human mobility. Specifically, we first exploit a probabilistic propagation-based approach to create a set of mobility graphs from periodic human mobility records. In these mobility graphs, the static POIs are regarded as vertexes, the dynamic mobility connectivities between POI pairs are regarded as edges, and the edge weights periodically evolve over time. A collective deep auto-encoder method is then developed to collaboratively learn the embeddings of POIs from multiple spatial-temporal mobility graphs. In addition, we develop a Unsupervised Graph based Weighted Aggregation method to align and aggregate the POI embeddings into the representation of the community structures. We apply the proposed embedding framework to two applications (i.e., spotting vibrant communities and predicting housing price return rates) to evaluate the performance of our proposed method. Extensive experimental results on real-world urban communities and human mobility data demonstrate the effectiveness of the proposed collective embedding framework.

The Ihara zeta function for infinite graphs

We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings, and percolation graphs. Making use of Connes’s noncommutative integration theory, we construct a zeta function and present a determinant formula for it. We further introduce a notion of weak convergence of measure graphs and show that our construction is compatible with it. The approximation of the Ihara zeta function via the normalized version on finite graphs in the sense of Benjamini and Schramm follows as a special case. Our framework not only unifies corresponding earlier results occurring in the literature. It likewise provides extensions to rich new classes of objects such as percolation graphs.

Nonlinear Partial Differential Equations on Graphs

One-dimensional metric graphs in two and three-dimensional spaces play an important role in emerging areas of modern science such as nano-technology, quantum physics, and biological networks. The workshop focused on the analysis of nonlinear partial differential equations on metric graphs, especially on the bifurcation and stability of nonlinear waves on complex graphs, on the justification of Kirchhoff boundary conditions, on spectral properties and the validity of amplitude equations for periodic graphs, and the existence of ground states for the NLS equation with and without potential.