Graph Theory and Additive Combinatorics: Exploring Structure and Randomness (1版)

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【簡介】 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. 【目錄】

Introduction to Graph Theory

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【簡介】 Graph theory is an area in discrete mathematics which studies configurations (called graphs) involving a set of vertices interconnected by edges. This book is intended as a general introduction to graph theory. The book builds on the verity that graph theory even at high school level is a subject that lends itself well to the development of mathematical reasoning and proof. This is an updated edition of two books already published with World Scientific, i.e., Introduction to Graph Theory: H3 Mathematics & Introduction to Graph Theory: Solutions Manual. The new edition includes solutions and hints to selected problems. This combination allows the book to be used as a textbook for undergraduate students. Professors can select unanswered problems for tutorials while students have solutions for reference. 【目錄】 Preface Notation Fundamental Concepts and Basic Results Graph Isomorphisms, Subgraphs, the Complement of a Graph Bipartite Graphs and Trees Vertex-colourings of Graphs Matchings in Bipartite Graphs Eulerian Multigraphs and Hamiltonian Graphs Digraphs and Tournaments Solutions of selected questions References Books Recommended Index

Applied Graph Theory

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書名：Applied Graph Theory 作者：Christopher Griffin 出版商：World Scientific 出版日期：2023-09-01 ISBN：9789811273100 內容簡介 DESCRIPTION This book serves as an introduction to graph theory and its applications. It is intended for a senior undergraduate course in graph theory but is also appropriate for beginning graduate students in science or engineering. The book presents a rigorous (proof-based) introduction to graph theory while also discussing applications of the results for solving real-world problems of interest. The book is divided into four parts. Part 1 covers the combinatorial aspects of graph theory including a discussion of common vocabulary, a discussion of vertex and edge cuts, Eulerian tours, Hamiltonian paths and a characterization of trees. This leads to Part 2, which discusses common combinatorial optimization problems. Spanning trees, shortest path problems and matroids are all discussed, as are maximum flow problems. Part 2 ends with a discussion of graph coloring and a proof of the NP-completeness of the coloring problem. Part 3 introduces the reader to algebraic graph theory, and focuses on Markov chains, centrality computation (e.g., eigenvector centrality and page rank), as well as spectral graph clustering and the graph Laplacian. Part 4 contains additional material on linear programming, which is used to provide an alternative analysis of the maximum flow problem. Two appendices containing prerequisite material on linear algebra and probability theory are also provided. Request Inspection Copy Sample Chapter(s) Preface Chapter 1: Introduction to Graph Theory Contents: Introduction to Graphs: Introduction to Graph Theory Degree Sequences and Subgraphs Walks, Cycles, Cuts, and Centrality Bipartite, Acyclic, and Eulerian Graphs Optimization in Graphs and NP-Completeness: Trees, Algorithms, and Matroids An Introduction to Network Flows and Combinatorial Optimization Coloring Some Algebraic Graph Theory: Algebraic Graph Theory with Abstract Algebra Algebraic Graph Theory with Linear Algebra Applications of Algebraic Graph Theory Linear Programming and Graph Theory: A Brief Introduction to Linear Programming Max Flow/Min Cut with Linear Programming Appendices: Fields, Vector Spaces, and Matrices A Brief Introduction to Probability Theory Readership: Advanced Undergraduate Students or Beginning Graduate Students in Mathematics (those who have taken a first course in proofs). Graduate Students in STEM who want a rigorous text on graph theory that also focuses on applications. This could be used as a secondary text in a physics course on Network Science, or potentially in a rigorous course in theoretical computer science or operations research with graph theory.

Graph Theory

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Graph Theory: An Introduction to Proofs, Algorithms, and Applications Graph theory is the study of interactions, conflicts, and connections. The relationship between collections of discrete objects can inform us about the overall network in which they reside, and graph theory can provide an avenue for analysis. This text, for the first undergraduate course, will explore major topics in graph theory from both a theoretical and applied viewpoint. Topics will progress from understanding basic terminology, to addressing computational questions, and finally ending with broad theoretical results. Examples and exercises will guide the reader through this progression, with particular care in strengthening proof techniques and written mathematical explanations. Current applications and exploratory exercises are provided to further the reader’s mathematical reasoning and understanding of the relevance of graph theory to the modern world. Features The first chapter introduces graph terminology, mathematical modeling using graphs, and a review of proof techniques featured throughout the book The second chapter investigates three major route problems: eulerian circuits, hamiltonian cycles, and shortest paths. The third chapter focuses entirely on trees – terminology, applications, and theory. Four additional chapters focus around a major graph concept: connectivity, matching, coloring, and planarity. Each chapter brings in a modern application or approach. Hints and Solutions to selected exercises provided at the back of the book. Author Karin R. Saoub is an Associate Professor of Mathematics at Roanoke College in Salem, Virginia. She earned her PhD in mathematics from Arizona State University and BA from Wellesley College. Her research focuses on graph coloring and on-line algorithms applied to tolerance graphs. She is also the author of A Tour Through Graph Theory, published by CRC Press.