scholarly journals Asymptotic Spectra of Large (Grid) Graphs with a Uniform Local Structure (Part I): Theory

2020 ◽  
Vol 88 (2) ◽  
pp. 409-454
Author(s):  
Andrea Adriani ◽  
Davide Bianchi ◽  
Stefano Serra-Capizzano
Keyword(s):  
Local Structure ◽  
Differential Operators ◽  
Spectral Gap ◽  
Large Graphs ◽  
Grid Graphs ◽  
Graphs And Networks ◽  
Adjacency Matrices ◽  
Distribution Type ◽  
Laplacian Operators ◽  
Large Grid

AbstractWe are mainly concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain $${\Omega} {\subset} \mathbb{R}^{d}, d \geq 1$$ Ω ⊂ R d , d ≥ 1 . When $$\Omega = [0, 1]$$ Ω = [ 0 , 1 ] , such graphs include the standard Toeplitz graphs and, for $$\Omega = [0, 1]^{d}$$ Ω = [ 0 , 1 ] d , the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and we show that we can associate to it a symbol $$\mathfrak{f}$$ f . The knowledge of the symbol and of its basic analytical features provides many information on the eigenvalue structure, of localization, spectral gap, clustering, and distribution type.Few generalizations are also considered in connection with the notion of generalized locally Toeplitz sequences and applications are discussed, stemming e.g. from the approximation of differential operators via numerical schemes. Nevertheless, more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks.

scholarly journals Isoscattering strings of concatenating graphs and networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Michał Ławniczak ◽  
Adam Sawicki ◽  
Małgorzata Białous ◽  
Leszek Sirko
Keyword(s):  
Trace Function ◽  
Quantum Graphs ◽  
Mathematical Approach ◽  
Large Graphs ◽  
Graphs And Networks ◽  
Scattering Matrices ◽  
Theoretical Predictions ◽  
Infinite Strings ◽  
Microwave Networks ◽  
Insight Into

AbstractWe identify and investigate isoscattering strings of concatenating quantum graphs possessing n units and 2n infinite external leads. We give an insight into the principles of designing large graphs and networks for which the isoscattering properties are preserved for $$n \rightarrow \infty $$ n → ∞ . The theoretical predictions are confirmed experimentally using $$n=2$$ n = 2 units, four-leads microwave networks. In an experimental and mathematical approach our work goes beyond prior results by demonstrating that using a trace function one can address the unsettled until now problem of whether scattering properties of open complex graphs and networks with many external leads are uniquely connected to their shapes. The application of the trace function reduces the number of required entries to the $$2n \times 2n $$ 2 n × 2 n scattering matrices $${\hat{S}}$$ S ^ of the systems to 2n diagonal elements, while the old measures of isoscattering require all $$(2n)^2$$ ( 2 n ) 2 entries. The studied problem generalizes a famous question of Mark Kac “Can one hear the shape of a drum?”, originally posed in the case of isospectral dissipationless systems, to the case of infinite strings of open graphs and networks.

scholarly journals -BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS

2017 ◽  
Vol 18 (3) ◽  
pp. 531-559 ◽  
Author(s):  
Julio Delgado ◽  
Michael Ruzhansky
Keyword(s):  
Phase Space ◽  
Vector Fields ◽  
Differential Operators ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Pseudo Differential Operators ◽  
Different Types ◽  
Unitary Dual ◽  
Laplacian Operators ◽  
Noncommutative Analogue

Given a compact Lie group$G$, in this paper we establish$L^{p}$-bounds for pseudo-differential operators in$L^{p}(G)$. The criteria here are given in terms of the concept of matrix symbols defined on the noncommutative analogue of the phase space$G\times \widehat{G}$, where$\widehat{G}$is the unitary dual of$G$. We obtain two different types of$L^{p}$bounds: first for finite regularity symbols and second for smooth symbols. The conditions for smooth symbols are formulated using$\mathscr{S}_{\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF}}^{m}(G)$classes which are a suitable extension of the well-known$(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF})$ones on the Euclidean space. The results herein extend classical$L^{p}$bounds established by C. Fefferman on$\mathbb{R}^{n}$. While Fefferman’s results have immediate consequences on general manifolds for$\unicode[STIX]{x1D70C}>\max \{\unicode[STIX]{x1D6FF},1-\unicode[STIX]{x1D6FF}\}$, our results do not require the condition$\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$. Moreover, one of our results also does not require$\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$. Examples are given for the case of$\text{SU}(2)\cong \mathbb{S}^{3}$and vector fields/sub-Laplacian operators when operators in the classes$\mathscr{S}_{0,0}^{m}$and$\mathscr{S}_{\frac{1}{2},0}^{m}$naturally appear, and where conditions$\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$and$\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$fail, respectively.

scholarly journals Globally Optimal Dense and Sparse Spanning Trees, and their Applications

2020 ◽  
Vol 8 (2) ◽  
pp. 328-345
Author(s):  
Mustafa Ozen ◽  
Goran Lesaja ◽  
Hua Wang
Keyword(s):  
Spanning Trees ◽  
Optimization Problems ◽  
Large Graphs ◽  
Classic Problem ◽  
Graphs And Networks ◽  
Vertex Degrees ◽  
New Feature ◽  
First Time ◽  
Sum Of Distances ◽  
Additional Constraints

Finding spanning trees under various constraints is a classic problem with applications in many fields. Recently, a novel notion of dense ( sparse ) tree, and in particular spanning tree (DST and SST respectively), is introduced as the structure that have a large (small) number of subtrees, or small (large) sum of distances between vertices. We show that finding DST and SST reduces to solving the discrete optimization problems. New and efficient approaches to find such spanning trees is achieved by imposing certain conditions on the vertex degrees which are then used to define an objective function that is minimized over all spanning trees of the graph under consideration. Solving this minimization problem exactly may be prohibitively time consuming for large graphs. Hence, we propose to use genetic algorithm (GA) which is one of well known metaheuristics methods to solve DST and SST approximately. As far as we are aware this is the first time GA has been used in this context.We also demonstrate on a number of applications that GA approach is well suited for these types of problems both in computational efficiency and accuracy of the approximate solution. Furthermore, we improve the efficiency of the proposed method by using Kruskal s algorithm in combination with GA. The application of our methods to several practical large graphs and networks is presented. Computational results show that they perform faster than previously proposed heuristic methods and produce more accurate solutions. Furthermore, the new feature of the proposed approach is that it can be applied recursively to sub-trees or spanning trees with additional constraints in order to further investigate the graphical properties of the graph and/or network. The application of this methodology on the gene network of a cancer cell led to isolating key genes in a network that were not obvious from previous studies.

scholarly journals Degeneration at E2 of certain spectral sequences

2016 ◽  
Vol 27 (14) ◽  
pp. 1650111 ◽  
Author(s):  
Dan Popovici
Keyword(s):  
Spectral Sequence ◽  
Differential Operators ◽  
Spectral Gap ◽  
Sufficient Conditions ◽  
Main Idea ◽  
Second Step ◽  
Compact Complex Manifold ◽  
Frölicher Spectral Sequence ◽  
Foliated Structure ◽  
Laplace Type

We propose a Hodge theory for the spaces [Formula: see text] featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold [Formula: see text] or in the spectral sequence associated with a pair [Formula: see text] of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on [Formula: see text] whose kernel in every bidegree [Formula: see text] is isomorphic to [Formula: see text] in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it involves a harmonic projection, although it depends on certain differential operators. We then use this Hodge isomorphism for [Formula: see text] to give sufficient conditions for the degeneration at [Formula: see text] of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on [Formula: see text]. For example, in the Frölicher case, we prove degeneration at [Formula: see text] if there exists an SKT metric [Formula: see text] (i.e. such that [Formula: see text]) whose torsion is small compared to the spectral gap of the elliptic operator [Formula: see text] defined by [Formula: see text]. In the foliated case, we obtain degeneration at [Formula: see text] under a hypothesis involving the Laplacians [Formula: see text] and [Formula: see text] associated with the splitting [Formula: see text] induced by the foliated structure.

scholarly journals On the number of eigenvalues in the spectral gap of a Dirac system

1986 ◽  
Vol 29 (3) ◽  
pp. 367-378 ◽  
Author(s):  
D. B. Hinton ◽  
A. B. Mingarelli ◽  
T. T. Read ◽  
J. K. Shaw
Keyword(s):  
Differential Operators ◽  
Spectral Gap ◽  
Equivalence Classes ◽  
Value Functions ◽  
Complex Vector ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Integrable Functions ◽  
The One ◽  
Image Position

We consider the one-dimensional operator,on 0<x<∞ with. The coefficientsp,V1andV2are assumed to be real, locally Lebesgue integrable functions;c1andc2are positive numbers. The operatorLacts in the Hilbert spaceHof all equivalence classes of complex vector-value functionssuch that.Lhas domainD(L)consisting of ally∈Hsuch thatyis locally absolutely continuous andLy∈H; thus in the language of differential operatorsLis a maximal operator. Associated withLis the minimal operatorL0defined as the closure ofwhereis the restriction ofLto the functions with compact support in (0,∞).

scholarly journals Characterization of Eigenvalues in Spectral Gap for Singular Differential Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Zhaowen Zheng ◽  
Wenju Zhang
Keyword(s):  
Positive Integer ◽  
Spectral Properties ◽  
Differential Operators ◽  
Spectral Gap ◽  
Deficiency Index ◽  
Minimal Operator ◽  
Adjoint Extension

The spectral properties fornorder differential operators are considered. When given a spectral gap(a,b)of the minimal operatorT0with deficiency indexr, arbitrarympointsβi  (i=1,2,…,m)in(a,b), and a positive integer functionpsuch that∑i=1mp(βi)≤r,T0has a self-adjoint extensionT̃such that eachβi  (i=1,2,…,m)is an eigenvalue ofT̃with multiplicity at leastp(βi).

scholarly journals Optimization Models for Efficient (t, r) Broadcast Domination in Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1028
Author(s):  
Poompol Buathong ◽  
Tipaluck Krityakierne
Keyword(s):  
Biological Networks ◽  
Domination Number ◽  
Real Life ◽  
Point Of View ◽  
Optimization Approach ◽  
Grid Graphs ◽  
Theoretical Approaches ◽  
Graphs And Networks ◽  
Domination In Graphs ◽  
Distance Domination

Known to be NP-complete, domination number problems in graphs and networks arise in many real-life applications, ranging from the design of wireless sensor networks and biological networks to social networks. Initially introduced by Blessing et al., the (t,r) broadcast domination number is a generalization of the distance domination number. While some theoretical approaches have been addressed for small values of t,r in the literature; in this work, we propose an approach from an optimization point of view. First, the (t,r) broadcast domination number is formulated and solved using linear programming. The efficient broadcast, whose wasted signals are minimized, is then found by a genetic algorithm modified for a binary encoding. The developed method is illustrated with several grid graphs: regular, slant, and king’s grid graphs. The obtained computational results show that the method is able to find the exact (t,r) broadcast domination number, and locate an efficient broadcasting configuration for larger values of t,r than what can be provided from a theoretical basis. The proposed optimization approach thus helps overcome the limitations of existing theoretical approaches in graph theory.

scholarly journals n-Laplacians on Metric Graphs and Almost Periodic Functions: I

2020 ◽  
Author(s):  
Pavel Kurasov ◽  
Jacob Muller
Keyword(s):  
Trigonometric Polynomial ◽  
Differential Operators ◽  
Secular Equation ◽  
Spectral Asymptotics ◽  
Almost Periodic Functions ◽  
General Interest ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Metric Graphs ◽  
Laplacian Operators

AbstractThe spectra of n-Laplacian operators $$(-\Delta )^n$$ ( - Δ ) n on finite metric graphs are studied. An effective secular equation is derived and the spectral asymptotics are analysed, exploiting the fact that the secular function is close to a trigonometric polynomial. The notion of the quasispectrum is introduced, and its uniqueness is proved using the theory of almost periodic functions. To achieve this, new results concerning roots of functions close to almost periodic functions are proved. The results obtained on almost periodic functions are of general interest outside the theory of differential operators.

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