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Using the dichotomy of structure and pseudorandomness as a central theme, this accessible text provides a modern introduction to extremal graph theory and additive combinatorics. Readers will explore central results in additive combinatorics-notably the cornerstone theorems of Roth, Szemerédi, Freiman, and Green-Tao-and will gain additional insights into these ideas through graph theoretic perspectives. Topics discussed include the Turán problem, Szemerédi's graph regularity method, pseudorandom graphs, graph limits, graph homomorphism inequalities, Fourier analysis in additive combinatorics, the structure of set addition, and the sum-product problem. Important combinatorial, graph theoretic, analytic, Fourier, algebraic, and geometric methods are highlighted. Students will appreciate the chapter summaries, many figures and exercises, and freely available lecture videos on MIT OpenCourseWare. Meant as an introduction for students and researchers studying combinatorics, theoretical computer science, analysis, probability, and number theory, the text assumes only basic familiarity with abstract algebra, analysis, and linear algebra.
Book Synopsis Graph Theory and Additive Combinatorics by : Yufei Zhao
Download or read book Graph Theory and Additive Combinatorics written by Yufei Zhao and published by Cambridge University Press. This book was released on 2023-07-31 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: Using the dichotomy of structure and pseudorandomness as a central theme, this accessible text provides a modern introduction to extremal graph theory and additive combinatorics. Readers will explore central results in additive combinatorics-notably the cornerstone theorems of Roth, Szemerédi, Freiman, and Green-Tao-and will gain additional insights into these ideas through graph theoretic perspectives. Topics discussed include the Turán problem, Szemerédi's graph regularity method, pseudorandom graphs, graph limits, graph homomorphism inequalities, Fourier analysis in additive combinatorics, the structure of set addition, and the sum-product problem. Important combinatorial, graph theoretic, analytic, Fourier, algebraic, and geometric methods are highlighted. Students will appreciate the chapter summaries, many figures and exercises, and freely available lecture videos on MIT OpenCourseWare. Meant as an introduction for students and researchers studying combinatorics, theoretical computer science, analysis, probability, and number theory, the text assumes only basic familiarity with abstract algebra, analysis, and linear algebra.
Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.
Book Synopsis Additive Combinatorics by : Terence Tao
Download or read book Additive Combinatorics written by Terence Tao and published by Cambridge University Press. This book was released on 2006-09-14 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt: Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.
Additive combinatorics is a relatively recent term coined to comprehend the developments of the more classical additive number theory, mainly focussed on problems related to the addition of integers. Some classical problems like the Waring problem on the sum of k-th powers or the Goldbach conjecture are genuine examples of the original questions addressed in the area. One of the features of contemporary additive combinatorics is the interplay of a great variety of mathematical techniques, including combinatorics, harmonic analysis, convex geometry, graph theory, probability theory, algebraic geometry or ergodic theory. This book gathers the contributions of many of the leading researchers in the area and is divided into three parts. The two first parts correspond to the material of the main courses delivered, Additive combinatorics and non-unique factorizations, by Alfred Geroldinger, and Sumsets and structure, by Imre Z. Ruzsa. The third part collects the notes of most of the seminars which accompanied the main courses, and which cover a reasonably large part of the methods, techniques and problems of contemporary additive combinatorics.
Book Synopsis Combinatorial Number Theory and Additive Group Theory by : Alfred Geroldinger
Download or read book Combinatorial Number Theory and Additive Group Theory written by Alfred Geroldinger and published by Springer Science & Business Media. This book was released on 2009-06-04 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: Additive combinatorics is a relatively recent term coined to comprehend the developments of the more classical additive number theory, mainly focussed on problems related to the addition of integers. Some classical problems like the Waring problem on the sum of k-th powers or the Goldbach conjecture are genuine examples of the original questions addressed in the area. One of the features of contemporary additive combinatorics is the interplay of a great variety of mathematical techniques, including combinatorics, harmonic analysis, convex geometry, graph theory, probability theory, algebraic geometry or ergodic theory. This book gathers the contributions of many of the leading researchers in the area and is divided into three parts. The two first parts correspond to the material of the main courses delivered, Additive combinatorics and non-unique factorizations, by Alfred Geroldinger, and Sumsets and structure, by Imre Z. Ruzsa. The third part collects the notes of most of the seminars which accompanied the main courses, and which cover a reasonably large part of the methods, techniques and problems of contemporary additive combinatorics.
Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. To develop a mathematical theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs, which has emerged over the last decade. The theory has rich connections with other approaches to the study of large networks, such as ``property testing'' in computer science and regularity partition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connections with other parts of mathematics (classical and non-classical, like probability theory, measure theory, tensor algebras, and semidefinite optimization). This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the theory of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits. This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future. --Persi Diaconis, Stanford University This book is a comprehensive study of the active topic of graph limits and an updated account of its present status. It is a beautiful volume written by an outstanding mathematician who is also a great expositor. --Noga Alon, Tel Aviv University, Israel Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovasz's book exemplifies this phenomenon. This book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory. --Terence Tao, University of California, Los Angeles, CA Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovasz's position as the main architect of this rapidly developing theory. The book is a must for combinatorialists, network theorists, and theoretical computer scientists alike. --Bela Bollobas, Cambridge University, UK
Book Synopsis Large Networks and Graph Limits by : László Lovász
Download or read book Large Networks and Graph Limits written by László Lovász and published by American Mathematical Soc.. This book was released on 2012 with total page 495 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. To develop a mathematical theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs, which has emerged over the last decade. The theory has rich connections with other approaches to the study of large networks, such as ``property testing'' in computer science and regularity partition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connections with other parts of mathematics (classical and non-classical, like probability theory, measure theory, tensor algebras, and semidefinite optimization). This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the theory of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits. This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future. --Persi Diaconis, Stanford University This book is a comprehensive study of the active topic of graph limits and an updated account of its present status. It is a beautiful volume written by an outstanding mathematician who is also a great expositor. --Noga Alon, Tel Aviv University, Israel Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovasz's book exemplifies this phenomenon. This book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory. --Terence Tao, University of California, Los Angeles, CA Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovasz's position as the main architect of this rapidly developing theory. The book is a must for combinatorialists, network theorists, and theoretical computer scientists alike. --Bela Bollobas, Cambridge University, UK
A basic problem for the interconnection of communications media is to design interconnection networks for specific needs. For example, to minimize delay and to maximize reliability, networks are required that have minimum diameter and maximum connectivity under certain conditions. The book provides a recent solution to this problem. The subject of all five chapters is the interconnection problem. The first two chapters deal with Cayley digraphs which are candidates for networks of maximum connectivity with given degree and number of nodes. Chapter 3 addresses Bruijn digraphs, Kautz digraphs, and their generalizations, which are candidates for networks of minimum diameter and maximum connectivity with given degree and number of nodes. Chapter 4 studies double loop networks, and Chapter 5 considers broadcasting and the Gossiping problem. All the chapters emphasize the combinatorial aspects of network theory. Audience: A vital reference for graduate students and researchers in applied mathematics and theoretical computer science.
Book Synopsis Combinatorial Network Theory by : Ding-Zhu Du
Download or read book Combinatorial Network Theory written by Ding-Zhu Du and published by Springer. This book was released on 2010-12-03 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: A basic problem for the interconnection of communications media is to design interconnection networks for specific needs. For example, to minimize delay and to maximize reliability, networks are required that have minimum diameter and maximum connectivity under certain conditions. The book provides a recent solution to this problem. The subject of all five chapters is the interconnection problem. The first two chapters deal with Cayley digraphs which are candidates for networks of maximum connectivity with given degree and number of nodes. Chapter 3 addresses Bruijn digraphs, Kautz digraphs, and their generalizations, which are candidates for networks of minimum diameter and maximum connectivity with given degree and number of nodes. Chapter 4 studies double loop networks, and Chapter 5 considers broadcasting and the Gossiping problem. All the chapters emphasize the combinatorial aspects of network theory. Audience: A vital reference for graduate students and researchers in applied mathematics and theoretical computer science.
An introductory text covering classical and modern developments in graph theory and additive combinatorics, based on Zhao's MIT course.
Book Synopsis Graph Theory and Additive Combinatorics by : Yufei Zhao
Download or read book Graph Theory and Additive Combinatorics written by Yufei Zhao and published by Cambridge University Press. This book was released on 2023-07-31 with total page 335 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introductory text covering classical and modern developments in graph theory and additive combinatorics, based on Zhao's MIT course.
Applications of Graph Theory gives an introduction on the subject of graph theory and the applications related to it. It explains the various computational complexities and the methodologies to solve the problems using NP/P graphs. Also discussed in the book are the theoretical applications of the graphs, the role of graphs in education, the application of graph theory in the recognition of language and the various special classes into which graphs and its applications are classified. The book also gives some conclusive remarks on the subject.
Book Synopsis Applications of Graph Theory by : Ivan Stanimirovic
Download or read book Applications of Graph Theory written by Ivan Stanimirovic and published by Arcler Press. This book was released on 2019-11 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Applications of Graph Theory gives an introduction on the subject of graph theory and the applications related to it. It explains the various computational complexities and the methodologies to solve the problems using NP/P graphs. Also discussed in the book are the theoretical applications of the graphs, the role of graphs in education, the application of graph theory in the recognition of language and the various special classes into which graphs and its applications are classified. The book also gives some conclusive remarks on the subject.
This volume explains the general theory of hypergraphs and presents in-depth coverage of fundamental and advanced topics: fractional matching, fractional coloring, fractional edge coloring, fractional arboricity via matroid methods, fractional isomorphism, and more. 1997 edition.
Book Synopsis Fractional Graph Theory by : Edward R. Scheinerman
Download or read book Fractional Graph Theory written by Edward R. Scheinerman and published by Courier Corporation. This book was released on 2013-04-29 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume explains the general theory of hypergraphs and presents in-depth coverage of fundamental and advanced topics: fractional matching, fractional coloring, fractional edge coloring, fractional arboricity via matroid methods, fractional isomorphism, and more. 1997 edition.
Book Synopsis Results in Extremal Graph Theory, Ramsey Theory and Additive Combinatorics by : Oliver Janzer
Download or read book Results in Extremal Graph Theory, Ramsey Theory and Additive Combinatorics written by Oliver Janzer and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
One of the Erd\H{o}s-like cornerstones in incidence geometry from which many other results follow is the celebrated Szemer\'edi-Trotter Theorem which states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this thesis, we study the effect of forbidding grids and short even cycles on the incidence graphs of point-line arrangements in the plane. Let \(A\) and \(B\) be two disjoint finite sets of points in the plane such that their union contains no three points on a line. We say that \(A\) \emph{avoids} \(B\) if no straight line determined by a pair of points in \(A\) intersects the convex hull of $B.$ $A$ and \(B\) are called mutually avoiding if \(A\) avoids \(B\) and \(B\) avoids \(A .\) Aronov et al. showed that any set of \(n\) points in general position in the plane contains a pair of mutually avoiding sets, each of size at least \(\Omega(\sqrt{n})\). Moreover, they proved that any set of \(n\) points in general position in \(\mathbb{R}^{d}\) contains a pair of mutually avoiding sets, each of size at least \(\Omega\left(n^{\frac{1}{d^{2}-d+1}}\right)\). In this thesis, we give a generalized version of mutually avoiding set theorem in the plane. Given an algebraic structure \(R\) and a subset \(A \subset R,\) define the sum set and the product set of \(A\) to be \(A+A=\{a+b: a, b \in A\}\) and \(A \cdot A=\{a \cdot b: a, b \in A\}\) respectively. Showing under what conditions at least one of \(|A+A|\) or \(|A \cdot A|\) is large has a long history of study that continues to the present day. By employing recent developments on the energy of polynomials over finite fields, we give the best-known lower bounds on $\max\{|A+A|, |f(A,A)|\}$, when $A$ is a small subset of $ \mathbb{F}_p,$ and $f$ is a quadratic nondegenerate polynomial in $\mathbb{F}_p[x,y].$
Book Synopsis Connections Between Additive Combinatorics, Graph Theory, and Incidence Geometry by : Mozhgan Mirzaei
Download or read book Connections Between Additive Combinatorics, Graph Theory, and Incidence Geometry written by Mozhgan Mirzaei and published by . This book was released on 2020 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the Erd\H{o}s-like cornerstones in incidence geometry from which many other results follow is the celebrated Szemer\'edi-Trotter Theorem which states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this thesis, we study the effect of forbidding grids and short even cycles on the incidence graphs of point-line arrangements in the plane. Let \(A\) and \(B\) be two disjoint finite sets of points in the plane such that their union contains no three points on a line. We say that \(A\) \emph{avoids} \(B\) if no straight line determined by a pair of points in \(A\) intersects the convex hull of $B.$ $A$ and \(B\) are called mutually avoiding if \(A\) avoids \(B\) and \(B\) avoids \(A .\) Aronov et al. showed that any set of \(n\) points in general position in the plane contains a pair of mutually avoiding sets, each of size at least \(\Omega(\sqrt{n})\). Moreover, they proved that any set of \(n\) points in general position in \(\mathbb{R}^{d}\) contains a pair of mutually avoiding sets, each of size at least \(\Omega\left(n^{\frac{1}{d^{2}-d+1}}\right)\). In this thesis, we give a generalized version of mutually avoiding set theorem in the plane. Given an algebraic structure \(R\) and a subset \(A \subset R,\) define the sum set and the product set of \(A\) to be \(A+A=\{a+b: a, b \in A\}\) and \(A \cdot A=\{a \cdot b: a, b \in A\}\) respectively. Showing under what conditions at least one of \(|A+A|\) or \(|A \cdot A|\) is large has a long history of study that continues to the present day. By employing recent developments on the energy of polynomials over finite fields, we give the best-known lower bounds on $\max\{|A+A|, |f(A,A)|\}$, when $A$ is a small subset of $ \mathbb{F}_p,$ and $f$ is a quadratic nondegenerate polynomial in $\mathbb{F}_p[x,y].$
