Download or read book On Expander Graphs and Hypergraphs written by Thiradet Jiarasuksakun and published by . This book was released on 2006 with total page 58 pages. Available in PDF, EPUB and Kindle. Book excerpt: Expander graphs have been studied in various definitions and approaches. We show some relationships among these definitions. A clear criterion for optimality of expanders is available only in one approach, namely the spectral approach, where the optimal graphs are called Ramanujan graphs. Therefore many new discoveries are based on the second largest absolute values of eigenvalues of graphs. A special expander graph called a unique-neighbor expander as shown by Alon and Capalbo in 2002 have been presented and generalized. In order to achieve an easy explicit family of fixed degree good expander graphs, we make recursions with Reingold's, Vadhan's, and Wigderson's zig-zag product and a new operation. We also introduce the C 3-lift construction inspired by Bilu's and Linial's 2-lift in 2004 to generate a bigger expander without hurting much of the expansion property. Unfortunately, most techniques on graphs are not applicable when it comes to hypergraphs. We give a new definition to study the spectrum of an expander hypergraph so that the results can be compared to the case of graphs.
Download or read book An Introduction to Expander Graphs written by and published by . This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Expander Graphs written by Masoumeh Soleimani Amirshekari and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Hypergraphs written by C. Berge and published by Elsevier. This book was released on 1984-05-01 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graph Theory has proved to be an extremely useful tool for solving combinatorial problems in such diverse areas as Geometry, Algebra, Number Theory, Topology, Operations Research and Optimization. It is natural to attempt to generalise the concept of a graph, in order to attack additional combinatorial problems. The idea of looking at a family of sets from this standpoint took shape around 1960. In regarding each set as a ``generalised edge'' and in calling the family itself a ``hypergraph'', the initial idea was to try to extend certain classical results of Graph Theory such as the theorems of Turán and König. It was noticed that this generalisation often led to simplification; moreover, one single statement, sometimes remarkably simple, could unify several theorems on graphs. This book presents what seems to be the most significant work on hypergraphs.
Download or read book Expander Families and Cayley Graphs written by Mike Krebs and published by Oxford University Press. This book was released on 2011-09-30 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of expander graphs is a rapidly developing topic in mathematics and computer science, with applications to communication networks, error-correcting codes, cryptography, complexity theory, and much more. Expander Families and Cayley Graphs: A Beginner's Guide is a comprehensive introduction to expander graphs, designed to act as a bridge between classroom study and active research in the field of expanders. It equips those with little or no prior knowledge with the skills necessary to both comprehend current research articles and begin their own research. Central to this book are four invariants that measure the quality of a Cayley graph as a communications network-the isoperimetric constant, the second-largest eigenvalue, the diameter, and the Kazhdan constant. The book poses and answers three core questions: How do these invariants relate to one another? How do they relate to subgroups and quotients? What are their optimal values/growth rates? Chapters cover topics such as: · Graph spectra · A Cheeger-Buser-type inequality for regular graphs · Group quotients and graph coverings · Subgroups and Schreier generators · Ramanujan graphs and the Alon-Boppana theorem · The zig-zag product and its relation to semidirect products of groups · Representation theory and eigenvalues of Cayley graphs · Kazhdan constants The only introductory text on this topic suitable for both undergraduate and graduate students, Expander Families and Cayley Graphs requires only one course in linear algebra and one in group theory. No background in graph theory or representation theory is assumed. Examples and practice problems with varying complexity are included, along with detailed notes on research articles that have appeared in the literature. Many chapters end with suggested research topics that are ideal for student projects.
Download or read book Hypergraphs and Automorphic Forms written by Cristina M. Ballantine and published by . This book was released on 1998 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the late '80s, A. Lubotzky, R. Phillips and P. Samak [22] used Deligne's proof of Ramanujan's conjecture and the Jacquet-Langlands correspondence for cuspidal representations of 'GL'(2) to construct a class of Ramanujan graphs which are the best known expander graphs. Their graphs are Cayley graphs of the group 'PSL' 2,Z/qZ or 'PGL' 2,Z/qZ . Unfortunately, this strategy cannot be applied for groups in general because, in the general case, there is no equivalent of the Jacquet-Langlands correspondence. However, J. Rogawski has completely classified the representations of the unitary group in three variables in [25]. In this thesis we consider a form of 'U'(3) which at a place p is isomorphic to 'GL'3 Qp . We study a finite disconnected union of finite quotients of the building attached to the group 'GL'3 Qp . We view this object as a hypergraph and, using the classification of automorphic representations of the group 'U'(3) and Deligne's Theorem, we give an estimation of the spectrum of the adjacency matrix of its underlying graph. We show that the underlying graph is an expander graph with good expansion coefficient.
Download or read book G graphs and Expander Graphs written by Mohamad Badaoui and published by . This book was released on 2018 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Graphs and Hypergraphs written by Claude Berge and published by . This book was released on 1973 with total page 550 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Hypergraph Theory written by Alain Bretto and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to hypergraphs, its aim being to overcome the lack of recent manuscripts on this theory. In the literature hypergraphs have many other names such as set systems and families of sets. This work presents the theory of hypergraphs in its most original aspects, while also introducing and assessing the latest concepts on hypergraphs. The variety of topics, their originality and novelty are intended to help readers better understand the hypergraphs in all their diversity in order to perceive their value and power as mathematical tools. This book will be a great asset to upper-level undergraduate and graduate students in computer science and mathematics. It has been the subject of an annual Master's course for many years, making it also ideally suited to Master's students in computer science, mathematics, bioinformatics, engineering, chemistry, and many other fields. It will also benefit scientists, engineers and anyone else who wants to understand hypergraphs theory.
Download or read book Introduction to Global Variational Geometry written by Demeter Krupka and published by Elsevier. This book was released on 2009-06-15 with total page 544 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive introduction to modern global variational theory on fibred spaces. It is based on differentiation and integration theory of differential forms on smooth manifolds, and on the concepts of global analysis and geometry such as jet prolongations of manifolds, mappings, and Lie groups. The book will be invaluable for researchers and PhD students in differential geometry, global analysis, differential equations on manifolds, and mathematical physics, and for the readers who wish to undertake further rigorous study in this broad interdisciplinary field. Featured topics - Analysis on manifolds - Differential forms on jet spaces - Global variational functionals - Euler-Lagrange mapping - Helmholtz form and the inverse problem - Symmetries and the Noether’s theory of conservation laws - Regularity and the Hamilton theory - Variational sequences - Differential invariants and natural variational principles - First book on the geometric foundations of Lagrange structures - New ideas on global variational functionals - Complete proofs of all theorems - Exact treatment of variational principles in field theory, inc. general relativity - Basic structures and tools: global analysis, smooth manifolds, fibred spaces
Download or read book Expanding Graphs written by Joel Friedman and published by American Mathematical Soc.. This book was released on 1993-01-01 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book An Empirical Study of Expander Graphs and Graph Expansion written by Mark Allen Lotts and published by . This book was released on 2016 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: Expander graphs are commonly studied objects in computer science and mathematics that are found in the proofs of many important theorems. The vast majority of these theoretical uses of expanders rely on probabilistic statements of existence and do not grapple with the challenge of creating expander graphs or validating their expansion properties. In this paper, we will define expander graphs and describe different ways their expansion can be measured. We will discuss applications of expander graphs and provide empirical evidence of how they can be used in practice. We will also outline the difficulties of computing exact expansion rates and the hardness of estimating these rates, relating these problems to well-known results and conjectures in complexity theory. Using our own implementation of graph creation and verification algorithms, we will gain an empirical understanding of expander graphs, utilizing high-performance computing resources and repurposing well-known statistical methods to analyze expansion. We will show that, given an arbitrary graph, its potential to be used as an expander can be measured and bounded by employing community detection algorithms that seek to maximize modularity.
Download or read book Expander Graphs and where to Find Them written by Ana Khukhro and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Graphs are mathematical objects composed of a collection of “dots” called vertices, some of which are joined by lines called edges. Graphs are ideal for visually representing relations between things, and mathematical properties of graphs can provide an insight into real-life phenomena. One interesting property is how connected a graph is, in the sense of how easy it is to move between the vertices along the edges. The topic dealt with here is the construction of particularly well-connected graphs, and whether or not such graphs can happily exist in worlds similar to ours.
Download or read book A Proof of Alon s Second Eigenvalue Conjecture and Related Problems written by Joel Friedman and published by American Mathematical Soc.. This book was released on 2008 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: A $d$-regular graph has largest or first (adjacency matrix) eigenvalue $\lambda_1=d$. Consider for an even $d\ge 4$, a random $d$-regular graph model formed from $d/2$ uniform, independent permutations on $\{1,\ldots,n\}$. The author shows that for any $\epsilon>0$ all eigenvalues aside from $\lambda_1=d$ are bounded by $2\sqrt{d-1}\;+\epsilon$ with probability $1-O(n^{-\tau})$, where $\tau=\lceil \bigl(\sqrt{d-1}\;+1\bigr)/2 \rceil-1$. He also shows that this probability is at most $1-c/n^{\tau'}$, for a constant $c$ and a $\tau'$ that is either $\tau$ or $\tau+1$ (``more often'' $\tau$ than $\tau+1$). He proves related theorems for other models of random graphs, including models with $d$ odd.
Download or read book The Evolution of Expander Graphs written by David Y. Xiao and published by . This book was released on 2003 with total page 62 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Recent Trends in Combinatorics written by Andrew Beveridge and published by Springer. This book was released on 2016-04-12 with total page 775 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents some of the research topics discussed at the 2014-2015 Annual Thematic Program Discrete Structures: Analysis and Applications at the Institute for Mathematics and its Applications during Fall 2014, when combinatorics was the focus. Leading experts have written surveys of research problems, making state of the art results more conveniently and widely available. The three-part structure of the volume reflects the three workshops held during Fall 2014. In the first part, topics on extremal and probabilistic combinatorics are presented; part two focuses on additive and analytic combinatorics; and part three presents topics in geometric and enumerative combinatorics. This book will be of use to those who research combinatorics directly or apply combinatorial methods to other fields.
Download or read book Expander Graphs written by Nabil Kahale and published by . This book was released on 1993 with total page 79 pages. Available in PDF, EPUB and Kindle. Book excerpt:
