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Relevant bibliographies by topics / Farey graph

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Academic literature on the topic 'Farey graph'

Author: Grafiati

Published: 11 January 2025

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Journal articles on the topic "Farey graph"

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Kurkofka, Jan. "Ubiquity and the Farey graph." European Journal of Combinatorics 95 (June 2021): 103326. http://dx.doi.org/10.1016/j.ejc.2021.103326.

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DEMİR, Bilal, and Mustafa KARATAŞ. "Farey graph and rational fixed points of the extended modular group." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 71, no. 4 (December 30, 2022): 1029–43. http://dx.doi.org/10.31801/cfsuasmas.1089480.

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Fixed points of matrices have many applications in various areas of science and mathematics. Extended modular group ¯¯¯¯ΓΓ¯ is the group of 2×22×2 matrices with integer entries and determinant ±1±1. There are strong connections between extended modular group, continued fractions and Farey graph. Farey graph is a graph with vertex set ^Q=Q∪{∞}Q^=Q∪{∞}. In this study, we consider the elements in ¯¯¯¯ΓΓ¯ that fix rationals. For a given rational number, we use its Farey neighbours to obtain the matrix representation of the element in $\overline{\Gamma}$ that fixes the given rational. Then we express such elements as words in terms of generators using the relations between the Farey graph and continued fractions. Finally we give the new block reduced form of these words which all blocks have Fibonacci numbers entries.

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Calero-Sanz, Jorge. "On the Degree Distribution of Haros Graphs." Mathematics 11, no. 1 (December 26, 2022): 92. http://dx.doi.org/10.3390/math11010092.

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Haros graphs are a graph-theoretical representation of real numbers in the unit interval. The degree distribution of the Haros graphs provides information regarding the topological structure and the associated real number. This article provides a comprehensive demonstration of a conjecture concerning the analytical formulation of the degree distribution. Specifically, a theorem outlines the relationship between Haros graphs, the corresponding continued fraction of its associated real number, and the subsequent symbolic paths in the Farey binary tree. Moreover, an expression that is continuous and piecewise linear in subintervals defined by Farey fractions can be derived from an additional conclusion for the degree distribution of Haros graphs.

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Taylor, Samuel J., and Alexander Zupan. "Products of Farey graphs are totally geodesic in the pants graph." Journal of Topology and Analysis 08, no. 02 (March 15, 2016): 287–311. http://dx.doi.org/10.1142/s1793525316500096.

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We show that for a surface [Formula: see text], the subgraph of the pants graph determined by fixing a collection of curves that cut [Formula: see text] into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made in [2] and has the implication that an embedded product of Farey graphs in any pants graph is totally geodesic. In addition, we show that a pants graph contains a convex [Formula: see text]-flat if and only if it contains an [Formula: see text]-quasi-flat.

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Liao, Yunhua, Yaoping Hou, and Xiaoling Shen. "Tutte polynomial of a small-world Farey graph." EPL (Europhysics Letters) 104, no. 3 (November 1, 2013): 38001. http://dx.doi.org/10.1209/0295-5075/104/38001.

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Kurkofka, Jan. "The Farey graph is uniquely determined by its connectivity." Journal of Combinatorial Theory, Series B 151 (November 2021): 223–34. http://dx.doi.org/10.1016/j.jctb.2021.06.006.

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Zhang, Zhongzhi, Bin Wu, and Yuan Lin. "Counting spanning trees in a small-world Farey graph." Physica A: Statistical Mechanics and its Applications 391, no. 11 (June 2012): 3342–49. http://dx.doi.org/10.1016/j.physa.2012.01.039.

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Wright, Alex. "Spheres in the curve graph and linear connectivity of the Gromov boundary." Communications of the American Mathematical Society 4, no. 12 (September 4, 2024): 548–77. http://dx.doi.org/10.1090/cams/38.

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We consider the curve graph in the cases where it is not a Farey graph, and show that its Gromov boundary is linearly connected. For a fixed center point c c and radius r r , we define the sphere of radius r r to be the induced subgraph on the set of vertices of distance r r from c c . We show that these spheres are always connected in high enough complexity, and prove a slightly weaker result for low complexity surfaces.

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Hockman, Meira. "The Farey octahedron graph, the Poincaré polyhedron theorem and Gaussian integer continued fractions." Annales mathématiques du Québec 44, no. 1 (April 22, 2019): 149–64. http://dx.doi.org/10.1007/s40316-019-00115-4.

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Liao, Yunhua, Mohamed Maama, and M. A. Aziz-Alaoui. "Optimal networks for exact controllability." International Journal of Modern Physics C 31, no. 10 (August 20, 2020): 2050144. http://dx.doi.org/10.1142/s0129183120501442.

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The exact controllability can be mapped to the problem of maximum algebraic multiplicity of all eigenvalues. In this paper, we focus on the exact controllability of deterministic complex networks. First, we explore the eigenvalues of two famous networks, i.e. the comb-of-comb network and the Farey graph. Due to their special structure, we find that the eigenvalues of each network are mutually distinct, showing that these two networks are optimal networks with respect to exact controllability. Second, we study how to optimize the exact controllability of a deterministic network. Based on the spectral graph theory, we find that reducing the order of duplicate sets or co-duplicate sets which are two special vertex subsets can decrease greatly the exact controllability. This result provides an answer to an open problem of Li et al. [X. F. Li, Z. M. Lu and H. Li, Int. J. Mod. Phys. C 26, 1550028 (2015)]. Finally, we discuss the relation between the topological structure and the multiplicity of two special eigenvalues and the computational complexity of our method.

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Dissertations / Theses on the topic "Farey graph"

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Kushwaha, Seema. "Study of continued fractions arising from subgraphs of the farey graph." Thesis, IIT Delhi, 2017. http://localhost:8080/xmlui/handle/12345678/7240.

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Kurkofka, Jan Verfasser], and Reinhard [Akademischer Betreuer] [Diestel. "Ends and tangles, stars and combs, minors and the Farey graph / Jan Kurkofka ; Betreuer: Reinhard Diestel." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2020. http://nbn-resolving.de/urn:nbn:de:gbv:18-106601.

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Kurkofka, Jan [Verfasser], and Reinhard [Akademischer Betreuer] Diestel. "Ends and tangles, stars and combs, minors and the Farey graph / Jan Kurkofka ; Betreuer: Reinhard Diestel." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2020. http://d-nb.info/1216998116/34.

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Leclere, Ludivine. "q-analogues des nombres réels et des matrices unimodulaires : aspects algébriques, combinatoires et analytiques." Electronic Thesis or Diss., Reims, 2024. http://www.theses.fr/2024REIMS019.

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Ce travail est consacré à l'étude des q-analogues de nombre réels. La q-déformation d'un nombre rationnel utilisée est une fraction rationnelle à coefficients entiers qui a été introduite par Sophie Morier-Genoud et Valentin Ovsienko en 2019. Il s'agit dans un premier temps de préciser les propriétés algébriques et de donner des interprétations combinatoires des q-rationnels. On manipule les différentes notions liées aux nombres rationnels: les fractions continues, les matrices de PSL(2,Z), les continuants d'Euler, les triangulations de polygones et le graphe de Farey, et leurs versions q-déformées. Les traces des q-matrices de PSL(2,Z) obtenues sont étudiées et interprétées dans le modèle combinatoire de triangulations d'anneaux. Dans un second temps, nous nous intéressons à la q-déformation des nombres irrationnels et plus particulièrement des irrationnels quadratiques. Nous obtenons une formule explicite permettant d'écrire les q-irrationnels quadratiques. On s'intéresse aux rayons de convergence des séries de Laurent obtenues à partir des q-déformations des nombres réels. Enfin, nous introduisons un second paramètre pour obtenir des (q,t)-déformations de nombres. Ces dernières sont étudiées sous un aspect combinatoire pour affiner les interprétations dans les modèles déjà présentes mais également dans les graphes en serpents
This work is devoted to the study of q-analogs of real numbers. The q-deformation of a rational number that we use is a rational funtion with integer coefficientswhich was introduced by Sophie Morier-Genoud and Valentin Ovsienko in 2019. The first step is to elaborate algebraic properties and to give combinatorial interpretations of the q-rationals. We use different notions linked to rational numbers: continued fractions, PSL(2,Z) matrices, Euler continuants, polygon's triangulations and the Farey graph, and their q-deformed versions.The traces of the q-matrices of PSL(2,Z) that we obtained are studied and interpreted in the combinatorial model of triangulation of annulus. In a second stage, we focus on the q-deformations of irrational real numbers, and more precisely on quadratic irrational real numbers. We obtain an explicit formula to describe q-deformed quadratic irrationals. We give estimate for the radii of convergence of the Laurent series obtained from the q-deformations of real numbers. Finally, we introduce a second parameter to obtain (q, t)-deformations of the rationals. The latter is studied in its combinatorial aspect, in the models already described but also in terms of snake graphs

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Collyer, Thomas P. A. "On generalised Farey graphs and applications to the curve complex." Thesis, University of Warwick, 2012. http://wrap.warwick.ac.uk/56823/.

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In the first part of the thesis, we introduce a family of simplicial complexes called tree complexes, which generalise the well-known Farey graph. We study numerous aspects of tree complexes. Firstly we show for a given dimension n, the tree complex K(n) is simplicially rigid. We then study the geodesics between a pair of given vertices x and y, giving a bound in terms of the distance between the vertices, and showing that there always exist a pair of vertices at a given distance which attains this bound. When n = 2, this bound is the ith Fibonacci number, where i is the distance between the two vertices. We next study the automorphism group of a tree complex, showing that it splits as a semi-direct product. Finally we study the coarse geometry of a tree complex, showing in particular that for n > 2 each tree complex is quasi-isometric to the simplicial tree T [infinity]. In the second part of the thesis, we study the curve complex of the five-holed sphere, C(S0,5), via subsurface projections to the four-holed sphere. We show that geodesically embedded pentagons, hexagons and heptagons are unique, up to the action of the mapping class group. We conjecture firstly that there are no larger geodesically embedded cycles in C(S0,5), and secondly that these methods might be used in a greatly simplified proof of the hyperbolicity of C(S0,5).

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Book chapters on the topic "Farey graph"

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Short, Ian, and Mairi Walker. "Even-Integer Continued Fractions and the Farey Tree." In Symmetries in Graphs, Maps, and Polytopes, 287–300. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-30451-9_15.

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Margalit, Dan. "Groups Acting on Trees." In Office Hours with a Geometric Group Theorist. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691158662.003.0003.

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This chapter considers groups acting on trees. It examines which groups act on which spaces and, if a group does act on a space, what it says about the group. These spaces are called trees—that is, connected graphs without cycles. A group action on a tree is free if no nontrivial element of the group preserves any vertex or any edge of the tree. The chapter first presents the theorem stating that: If a group G acts freely on a tree, then G is a free group. The condition that G is free is equivalent to the condition that G acts freely on a tree. The discussion then turns to the Farey tree and shows how to construct the Farey complex using the Farey graph. The chapter concludes by describing free and non-free actions on trees. Exercises and research projects are included.

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"Fare You Well." In Symmetry in Graphs, 468–73. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108553995.015.

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Jones, G. A., D. Singerman, and K. Wicks. "The modular group and generalized Farey graphs." In Groups St Andrews 1989, 316–38. Cambridge University Press, 1991. http://dx.doi.org/10.1017/cbo9780511661846.006.

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Conference papers on the topic "Farey graph"

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Xiao, Yuzhi, and Haixing Zhao. "Counting the number of spanning trees of generalization Farey graph." In 2013 9th International Conference on Natural Computation (ICNC). IEEE, 2013. http://dx.doi.org/10.1109/icnc.2013.6818271.

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de Fraysseix, Hubert, János Pach, and Richard Pollack. "Small sets supporting fary embeddings of planar graphs." In the twentieth annual ACM symposium. New York, New York, USA: ACM Press, 1988. http://dx.doi.org/10.1145/62212.62254.

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