Expander Families And Cayley Graphs
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Author |
: Mike Krebs |
Publisher |
: OUP USA |
Total Pages |
: 283 |
Release |
: 2011-10-21 |
ISBN-10 |
: 9780199767113 |
ISBN-13 |
: 0199767114 |
Rating |
: 4/5 (13 Downloads) |
Synopsis Expander Families and Cayley Graphs by : Mike Krebs
Expander families enjoy a wide range of applications in mathematics and computer science, and their study is a fascinating one in its own right. Expander Families and Cayley Graphs: A Beginner's Guide provides an introduction to the mathematical theory underlying these objects. The central notion in the book is that of expansion, which roughly means the quality of a graph as a communications network. Cayley graphs are certain graphs constructed from groups; they play a prominent role in the study of expander families. The isoperimetric constant, the second largest eigenvalue, the diameter, and the Kazhdan constant are four measures of the expansion quality of a Cayley graph. The book carefully develops these concepts, discussing their relationships to one another and to subgroups and quotients as well as their best-case growth rates. Topics include graph spectra (i.e., eigenvalues); a Cheeger-Buser-type inequality for regular graphs; group quotients and graph coverings; subgroups and Schreier generators; the Alon-Boppana theorem on the second largest eigenvalue of a regular graph; Ramanujan graphs; diameter estimates for Cayley graphs; the zig-zag product and its relation to semidirect products of groups; eigenvalues of Cayley graphs; Paley graphs; and Kazhdan constants. The book was written with undergraduate math majors in mind; indeed, several dozen of them field-tested it. The prerequisites are minimal: one course in linear algebra, and one course in group theory. No background in graph theory or representation theory is assumed; the book develops from scatch the required facts from these fields. The authors include not only overviews and quick capsule summaries of key concepts, but also details of potentially confusing lines of reasoning. The book contains ideas for student research projects (for capstone projects, REUs, etc.), exercises (both easy and hard), and extensive notes with references to the literature.
Author |
: |
Publisher |
: |
Total Pages |
: 0 |
Release |
: 2013 |
ISBN-10 |
: OCLC:878077935 |
ISBN-13 |
: |
Rating |
: 4/5 (35 Downloads) |
Synopsis Expander Families and Their Applications by :
This thesis is to provide a brief introduction to the mathematical theory of expander families and their applications. Throughout this paper we mainly deal with graphs with finitely many vertices and edges. The central notion of this thesis is that of expansion roughly meaning the quality of a graph as a communication network where the vertices represent entities and an edge connects two vertices. An ideal communication network is a large graph with large isoperimetric constant, meanwhile the number of the edges are not too large. Cayley graphs are constructed from groups and they play a very important role in expander families. The second largest eigenvalue, the isoperimetric constant, and the diameter are the main measures of the expansion quality of Cayley graphs. The "zig-zag product" provides a straightforward combinatorial method to construct expander families.
Author |
: Terence Tao |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 319 |
Release |
: 2015-04-16 |
ISBN-10 |
: 9781470421960 |
ISBN-13 |
: 1470421968 |
Rating |
: 4/5 (60 Downloads) |
Synopsis Expansion in Finite Simple Groups of Lie Type by : Terence Tao
Expander graphs are an important tool in theoretical computer science, geometric group theory, probability, and number theory. Furthermore, the techniques used to rigorously establish the expansion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combinatorics. This text focuses on the latter topic in the important case of Cayley graphs on finite groups of Lie type, developing tools such as Kazhdan's property (T), quasirandomness, product estimates, escape from subvarieties, and the Balog-Szemerédi-Gowers lemma. Applications to the affine sieve of Bourgain, Gamburd, and Sarnak are also given. The material is largely self-contained, with additional sections on the general theory of expanders, spectral theory, Lie theory, and the Lang-Weil bound, as well as numerous exercises and other optional material.
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: |
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: |
Total Pages |
: |
Release |
: |
ISBN-10 |
: 2856298982 |
ISBN-13 |
: 9782856298985 |
Rating |
: 4/5 (82 Downloads) |
Synopsis An Introduction to Expander Graphs by :
Author |
: Mohamad Badaoui |
Publisher |
: |
Total Pages |
: 0 |
Release |
: 2018 |
ISBN-10 |
: OCLC:1045424738 |
ISBN-13 |
: |
Rating |
: 4/5 (38 Downloads) |
Synopsis G-graphs and Expander Graphs by : Mohamad Badaoui
Applying algebraic and combinatorics techniques to solve graph problems leads to the birthof algebraic and combinatorial graph theory. This thesis deals mainly with a crossroads questbetween the two theories, that is, the problem of constructing infinite families of expandergraphs.From a combinatorial point of view, expander graphs are sparse graphs that have strongconnectivity properties. Expanders constructions have found extensive applications in bothpure and applied mathematics. Although expanders exist in great abundance, yet their explicitconstructions, which are very desirable for applications, are in general a hard task. Mostconstructions use deep algebraic and combinatorial approaches. Following the huge amountof research published in this direction, mainly through Cayley graphs and the Zig-Zagproduct, we choose to investigate this problem from a new perspective; namely by usingG-graphs theory and spectral hypergraph theory as well as some other techniques. G-graphsare like Cayley graphs defined from groups, but they correspond to an alternative construction.The reason that stands behind our choice is first a notable identifiable link between thesetwo classes of graphs that we prove. This relation is employed significantly to get many newresults. Another reason is the general form of G-graphs, that gives us the intuition that theymust have in many cases such as the relatively high connectivity property.The adopted methodology in this thesis leads to the identification of various approaches forconstructing an infinite family of expander graphs. The effectiveness of our techniques isillustrated by presenting new infinite expander families of Cayley and G-graphs on certaingroups. Also, since expanders stand in no single stem of graph theory, this brings us toinvestigate several closely related threads from a new angle. For instance, we obtain newresults concerning the computation of spectra of certain Cayley and G-graphs, and theconstruction of several new infinite classes of integral and Hamiltonian Cayley graphs.
Author |
: Alex Lubotzky |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 201 |
Release |
: 2010-02-17 |
ISBN-10 |
: 9783034603324 |
ISBN-13 |
: 3034603320 |
Rating |
: 4/5 (24 Downloads) |
Synopsis Discrete Groups, Expanding Graphs and Invariant Measures by : Alex Lubotzky
In the last ?fteen years two seemingly unrelated problems, one in computer science and the other in measure theory, were solved by amazingly similar techniques from representation theory and from analytic number theory. One problem is the - plicit construction of expanding graphs («expanders»). These are highly connected sparse graphs whose existence can be easily demonstrated but whose explicit c- struction turns out to be a dif?cult task. Since expanders serve as basic building blocks for various distributed networks, an explicit construction is highly des- able. The other problem is one posed by Ruziewicz about seventy years ago and studied by Banach [Ba]. It asks whether the Lebesgue measure is the only ?nitely additive measure of total measure one, de?ned on the Lebesgue subsets of the n-dimensional sphere and invariant under all rotations. The two problems seem, at ?rst glance, totally unrelated. It is therefore so- what surprising that both problems were solved using similar methods: initially, Kazhdan’s property (T) from representation theory of semi-simple Lie groups was applied in both cases to achieve partial results, and later on, both problems were solved using the (proved) Ramanujan conjecture from the theory of automorphic forms. The fact that representation theory and automorphic forms have anything to do with these problems is a surprise and a hint as well that the two questions are strongly related.
Author |
: Joel Friedman |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 162 |
Release |
: 1993-01-01 |
ISBN-10 |
: 0821870572 |
ISBN-13 |
: 9780821870570 |
Rating |
: 4/5 (72 Downloads) |
Synopsis Expanding Graphs by : Joel Friedman
This volume contains the proceedings of the DIMACS Workshop on Expander Graphs, held at Princeton University in May 1992. The subject of expanding graphs involves a number of different fields and gives rise to important connections among them. Many of these fields were represented at the workshop, including theoretical computer science, combinatorics, probability theory, representation theory, number theory, and differential geometry. With twenty-two talks and two open problem sessions, the workshop provided a unique opportunity for cross-fertilization of various areas. This volume will prove useful to mathematicians and computer scientists interested in current results in this area of research.
Author |
: Ravindra B. Bapat |
Publisher |
: Springer |
Total Pages |
: 197 |
Release |
: 2014-09-19 |
ISBN-10 |
: 9781447165699 |
ISBN-13 |
: 1447165691 |
Rating |
: 4/5 (99 Downloads) |
Synopsis Graphs and Matrices by : Ravindra B. Bapat
This new edition illustrates the power of linear algebra in the study of graphs. The emphasis on matrix techniques is greater than in other texts on algebraic graph theory. Important matrices associated with graphs (for example, incidence, adjacency and Laplacian matrices) are treated in detail. Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree, and its generalized version for arbitrary graphs, known as the resistance matrix. Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph. Such an extensive coverage of the subject area provides a welcome prompt for further exploration. The inclusion of exercises enables practical learning throughout the book. In the new edition, a new chapter is added on the line graph of a tree, while some results in Chapter 6 on Perron-Frobenius theory are reorganized. Whilst this book will be invaluable to students and researchers in graph theory and combinatorial matrix theory, it will also benefit readers in the sciences and engineering.
Author |
: Matthew C. H. Tointon |
Publisher |
: Cambridge University Press |
Total Pages |
: 220 |
Release |
: 2019-11-14 |
ISBN-10 |
: 9781108470735 |
ISBN-13 |
: 1108470734 |
Rating |
: 4/5 (35 Downloads) |
Synopsis Introduction to Approximate Groups by : Matthew C. H. Tointon
Provides a comprehensive exploration of the main concepts and techniques from the young, exciting field of approximate groups.
Author |
: Vadim Kaimanovich |
Publisher |
: Walter de Gruyter |
Total Pages |
: 545 |
Release |
: 2008-08-22 |
ISBN-10 |
: 9783110198089 |
ISBN-13 |
: 3110198088 |
Rating |
: 4/5 (89 Downloads) |
Synopsis Random Walks and Geometry by : Vadim Kaimanovich
Die jüngsten Entwicklungen zeigen, dass sich Wahrscheinlichkeitsverfahren zu einem sehr wirkungsvollen Werkzeug entwickelt haben, und das auf so unterschiedlichen Gebieten wie statistische Physik, dynamische Systeme, Riemann'sche Geometrie, Gruppentheorie, harmonische Analyse, Graphentheorie und Informatik.