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Pierreloic Meliot - One of the best experts on this subject based on the ideXlab platform.
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asymptotic representation theory and the spectrum of a random Geometric Graph on a compact lie group
Electronic Journal of Probability, 2019Co-Authors: Pierreloic MeliotAbstract:Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random Geometric Graph on $G$ is the random Graph $\Gamma _{\mathrm{geom} }(N,L)$ whose vertices are $N$ random points $g_1,\ldots ,g_N$ chosen under the Haar measure of $G$, and whose edges are the pairs $\{g_i,g_j\}$ with $d(g_i,g_j)\leq L$, $d$ being the distance associated to the standard Riemannian structure on $G$. In this paper, we describe the asymptotic behavior of the spectrum of the adjacency matrix of $\Gamma _{\mathrm{geom} }(N,L)$, when $N$ goes to infinity. 1. If $L$ is fixed and $N \to + \infty $ (Gaussian regime), then the largest eigenvalues of $\Gamma _{\mathrm{geom} }(N,L)$ converge after an appropriate renormalisation towards certain explicit linear combinations of values of Bessel functions. 2. If $L = O(N^{-\frac{1} {\dim G}})$ and $N \to +\infty $ (Poissonian regime), then the Geometric Graph $\Gamma _{\mathrm{geom} }(N,L)$ converges in the local Benjamini–Schramm sense, which implies the weak convergence in probability of the spectral measure of $\Gamma _{\mathrm{geom} }(N,L)$. In both situations, the representation theory of the group $G$ provides us with informations on the limit of the spectrum, and conversely, the computation of this limiting spectrum involves many classical tools from representation theory: Weyl’s character formula and the weight lattice in the Gaussian regime, and a degeneration of these objects in the Poissonian regime. The representation theoretic approach allows one to understand precisely how the degeneration from the Gaussian to the Poissonian regime occurs, and the article is written so as to highlight this degeneration phenomenon. In the Poissonian regime, this approach leads us to an algebraic conjecture on certain functionals of the irreducible representations of $G$.
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asymptotic representation theory and the spectrum of a random Geometric Graph on a compact lie group
arXiv: Probability, 2018Co-Authors: Pierreloic MeliotAbstract:Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random Geometric Graph on $G$ is the random Graph $\Gamma(N,L)$ whose vertices are $N$ random points $g_1,\ldots,g_N$ chosen under the Haar measure of $G$, and whose edges are the pairs $\{g_i,g_j\}$ with $d(g_i,g_j)\leq L$, $d$ being the distance associated to the standard Riemannian structure on $G$. In this paper, we describe the asymptotic behavior of the spectrum of the adjacency matrix of $\Gamma(N,L)$, when $N$ goes to infinity. If $L$ is fixed and $N \to + \infty$ (Gaussian regime), then the largest eigenvalues of $\Gamma(N,L)$ converge after an appropriate renormalisation towards certain explicit linear combinations of values of Bessel functions. If $L = O(N^{-\frac{1}{\dim G}})$ and $N \to +\infty$ (Poissonian regime), then the random Geometric Graph $\Gamma(N,L)$ converges in the local Benjamini-Schramm sense, which implies the weak convergence in probability of the spectral measure of $\Gamma(N,L)$. In this situation, we explain how to compute the first moments of the limiting spectral measure by using the asymptotic representation theory of the group $G$. The computation of the higher moments relies on a conjecture on certain functionals of the irreducible representations of $G$, which we state at the end of the paper.
A. M. Akhtyamov - One of the best experts on this subject based on the ideXlab platform.
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Degenerate Boundary Conditions for the Sturm-Liouville Problem on a Geometric Graph
Differential Equations, 2019Co-Authors: Victor Antonovich Sadovnichii, Ya. T. Sultanaev, A. M. AkhtyamovAbstract:We study the boundary conditions of the Sturm-Liouville problem posed on a star-shaped Geometric Graph consisting of three edges with a common vertex. We show that the Sturm-Liouville problem has no degenerate boundary conditions in the case of pairwise distinct edge lengths. However, if the edge lengths coincide and all potentials are the same, then the characteristic determinant of the Sturm-Liouville problem cannot be a nonzero constant and the set of Sturm-Liouville problems whose characteristic determinant is identically zero and whose spectrum accordingly coincides with the entire plane is infinite (a continuum). It is shown that, for one special case of the boundary conditions, this set consists of eighteen classes, each having from two to four arbitrary constants, rather than of two problems as in the case of the Sturm-Liouville problem on an interval.
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Inverse Sturm-Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph
Differential Equations, 2019Co-Authors: Victor Antonovich Sadovnichii, Ya. T. Sultanaev, A. M. AkhtyamovAbstract:The inverse Sturm-Liouville problem with nonseparated boundary conditions on a star-shaped Geometric Graph consisting of three edges with a common vertex is studied. It is shown that the Sturm-Liouville problem with general boundary conditions cannot be reconstructed uniquely from four spectra. A class of nonseparated boundary conditions is obtained for which two uniqueness theorems for the solution of the inverse Sturm-Liouville problem are proved. In the first theorem, the data used to reconstruct the Sturm-Liouville problem are the spectrum of the boundary value problem itself and the spectra of three auxiliary problems with separated boundary conditions. In the second theorem, instead of the spectrum of the problem itself, one only deals with five of its eigenvalues. It is shown that the Sturm-Liouville problem with these nonseparated boundary conditions can be reconstructed uniquely if three spectra of auxiliary problems and five eigenvalues of the problem itself are used as the reconstruction data. Examples of unique reconstruction of potentials and boundary conditions of the Sturm-Liouville problem posed on the Graph under study are given.
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on the uniqueness of the solution of the inverse sturm liouville problem with nonseparated boundary conditions on a Geometric Graph
Doklady Mathematics, 2018Co-Authors: V A Sadovnichy, Ya. T. Sultanaev, A. M. AkhtyamovAbstract:For the first time, the inverse Sturm–Liouville problem with nonseparated boundary conditions is studied on a star-shaped Geometric Graph with three edges. It is shown that the Sturm–Liouville problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville problem is proved. The spectrum of the boundary value problem itself and the spectra of three auxiliary problems are used as reconstruction data. It is also shown that the Sturm–Liouville problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the problem.
Xavier Perezgimenez - One of the best experts on this subject based on the ideXlab platform.
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on the relation between Graph distance and euclidean distance in random Geometric Graphs
Advances in Applied Probability, 2016Co-Authors: Josep Diaz, Dieter Mitsche, Guillem Perarnau, Xavier PerezgimenezAbstract:Given any two vertices u;v of a random Geometric Graph G (n;r), denote by dE(u;v) their Euclidean distance and by dG(u;v) their Graph distance. The problem of nding upper bounds on dG(u;v) conditional on dE(u;v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper, we improve the known upper bounds for values of r = !( p logn) (i.e. for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random Geometric Graphs. We also provide a lower bound on dG(u;v) conditional on dE(u;v).
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on the relation between Graph distance and euclidean distance in random Geometric Graphs
Advances in Applied Probability, 2016Co-Authors: Josep Diaz, Dieter Mitsche, Guillem Perarnau, Xavier PerezgimenezAbstract:Given any two vertices u, v of a random Geometric Graph G(n, r), denote by d E (u, v) their Euclidean distance and by d E (u, v) their Graph distance. The problem of finding upper bounds on d G (u, v) conditional on d E (u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=ω(√logn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random Geometric Graphs. We also provide a lower bound on d E (u, v) conditional on d E (u, v).
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on the relation between Graph distance and euclidean distance in random Geometric Graphs
arXiv: Discrete Mathematics, 2014Co-Authors: Josep Diaz, Dieter Mitsche, Guillem Perarnau, Xavier PerezgimenezAbstract:Given any two vertices u, v of a random Geometric Graph, denote by d_E(u,v) their Euclidean distance and by d_G(u,v) their Graph distance. The problem of finding upper bounds on d_G(u,v) in terms of d_E(u,v) has received a lot of attention in the literature. In this paper, we improve these upper bounds for values of r=omega(sqrt(log n)) (i.e. for r above the connectivity threshold). Our result also improves the best-known estimates on the diameter of random Geometric Graphs. We also provide a lower bound on d_G(u,v) in terms of d_E(u,v).
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disjoint hamilton cycles in the random Geometric Graph
Journal of Graph Theory, 2011Co-Authors: Tobias Muller, Xavier Perezgimenez, Nicholas C WormaldAbstract:We consider the standard random Geometric Graph process in which n vertices are placed at random on the unit square and edges are sequentially added in increasing order of edge-length. For fixed k⩾1, weprove that the first edge in the process that creates a k-connected Graph coincides a.a.s. with the first edge that causes the Graph to contain k/2 pairwise edge-disjoint Hamilton cycles (for even k), or (k−1)/2 Hamilton cycles plus one perfect matching, all of them pairwise edge-disjoint (for odd k). This proves and extends a conjecture of Krivelevich and M **image**ler. In the special case when k = 2, our result says that the first edge that makes the random Geometric Graph Hamiltonian is a.a.s. exactly the same one that gives 2-connectivity, which answers a question of Penrose. (This result appeared in three independent preprints, one of which was a precursor to this article.) We prove our results with lengths measured using the lp norm for any p>1, and we also extend our result to higher dimensions. © 2011 Wiley Periodicals, Inc. J Graph Theory 68:299-322, 2011 © 2011 Wiley Periodicals, Inc.
Michiel Smid - One of the best experts on this subject based on the ideXlab platform.
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fixed orientation equilateral triangle matching of point sets
Workshop on Algorithms and Computation, 2013Co-Authors: Jasine Babu, Ahmad Biniaz, Anil Maheshwari, Michiel SmidAbstract:Given a point set P and a class \(\mathcal{C}\) of Geometric objects, \(G_\mathcal{C}(P)\) is a Geometric Graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some \(C \in \mathcal{C}\) containing both p and q but no other points from P. We study G ∇ (P) Graphs where ∇ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these Graphs have been shown to be equivalent to half-Θ6 Graphs and TD-Delaunay Graphs.
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approximating the average stretch factor of Geometric Graphs
Journal of Computational Geometry, 2012Co-Authors: Siuwing Cheng, Christian Knauer, Stefan Langerman, Michiel SmidAbstract:Let G be a Geometric Graph whose vertex set S is a set of n points in ℝ d . The stretch factor of two distinct points p and q in S is the ratio of their shortest-path distance in G and their Euclidean distance. We consider the problem of approximating the average of the n choose 2 stretch factors determined by all pairs of points in S . We show that for paths, cycles, and trees, this average can be approximated, within a factor of 1+e, in O ( n polylog( n )) time. For plane Graphs in ℝ 2 , we present a (2+e)-approximation algorithm with running time O ( n 5/3 polylog( n )), and a (4+e)-approximation algorithm with running time O ( n 3/2 polylog( n )). Finally, we show that, for any tree in ℝ 2 , the exact average of the squares of the n choose 2 stretch factors can be computed in O ( n 11/6 ) time.
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approximating the average stretch factor of Geometric Graphs
International Symposium on Algorithms and Computation, 2010Co-Authors: Siuwing Cheng, Christian Knauer, Stefan Langerman, Michiel SmidAbstract:Let G be a Geometric Graph whose vertex set S is a set of n points in ℝ d . The stretch factor of two distinct points p and q in S is the ratio of their shortest-path distance in G and their Euclidean distance. We consider the problem of approximating the sum of all \(n \choose 2\) stretch factors determined by all pairs of points in S. We show that for paths, cycles, and trees, this sum can be approximated, within a factor of 1 + e, in O(n polylog(n)) time. For plane Graphs, we present a (2 + e)-approximation algorithm with running time O(n 5/3 polylog(n)), and a (4 + e)-approximation algorithm with running time O(n 3/2 polylog(n)).
Michael Schulz - One of the best experts on this subject based on the ideXlab platform.
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simultaneous Geometric Graph embeddings
Graph Drawing, 2007Co-Authors: Alejandro Estrellabalderrama, Elisabeth Gassner, Michael Junger, Merijam Percan, Marcus Schaefer, Michael SchulzAbstract:We consider the following problem known as simultaneous Geometric Graph embedding (SGE). Given a set of planar Graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each Graph the straight-line drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two Graphs the problem is NP-hard. We also show that the problem of computing the rectilinear crossing number of a Graph can be reduced to a simultaneous Geometric Graph embedding problem; this implies that placing SGE in NP will be hard, since the corresponding question for rectilinear crossing number is a long-standing open problem. However, rather like rectilinear crossing number, SGE can be decided in PSPACE.