ISBN 9781032475950
Published January 21, 2023
592 Pages 905 B/W Illustrations
Graph Theory and Its Applications, Third Edition is the latest edition of the international, bestselling textbook for undergraduate courses in graph theory, yet it is expansive enough to be used for graduate courses as well. The textbook takes a comprehensive, accessible approach to graph theory, integrating careful exposition of classical developments with emerging methods, models, and practical needs.
The authorsf unparalleled treatment is an ideal text for a two-semester course and a variety of one-semester classes, from an introductory one-semester course to courses slanted toward classical graph theory, operations research, data structures and algorithms, or algebra and topology.
Introduction to Graph Models
Graphs and Digraphs. Common Families of Graphs. Graph Modeling Applications. Walks and Distance. Paths, Cycles, and Trees. Vertex and Edge Attributes.
Structure and Representation
Graph Isomorphism. Automorphism and Symmetry. Subgraphs. Some Graph Operations. Tests for Non-Isomorphism. Matrix Representation. More Graph Operations.
Trees
Characterizations and Properties of Trees. Rooted Trees, Ordered Trees, and Binary Trees. Binary-Tree Traversals. Binary-Search Trees. Huffman Trees and Optimal Prefix Codes. Priority Trees. Counting Labeled Trees. Counting Binary Trees.
Spanning Trees
Tree Growing. Depth-First and Breadth-First Search. Minimum Spanning Trees and Shortest Paths. Applications of Depth-First Search. Cycles, Edge-Cuts, and Spanning Trees. Graphs and Vector Spaces. Matroids and the Greedy Algorithm.
Connectivity
Vertex and Edge-Connectivity. Constructing Reliable Networks. Max-Min Duality and Mengerfs Theorems. Block Decompositions.
Optimal Graph Traversals
Eulerian Trails and Tours. DeBruijn Sequences and Postman Problems. Hamiltonian Paths and Cycles. Gray Codes and Traveling Salesman Problems.
Planarity and Kuratowskifs Theorem
Planar Drawings and Some Basic Surfaces. Subdivision and Homeomorphism. Extending Planar Drawings. Kuratowskifs Theorem. Algebraic Tests for Planairty. Planarity Algorithm. Crossing Numbers and Thickness.
Graph Colorings
Vertex-Colorings. Map-Colorings. Edge-Colorings. Factorization.
Special Digraph Models
Directed Paths and Mutual Reachability. Digraphs as Models for Relations. Tournaments. Project Scheduling. Finding the Strong Components of a Digraph.
Network Flows and Applications
Flows and Cuts in Networks. Solving the Maximum-Flow Problem. Flows and Connectivity. Matchings, Transversals, and Vertex Covers.
Graph Colorings and Symmetry
Automorphisms of Simple Graphs. Equivalence Classes of Colorings.
Appendix
ISBN 9781032477251
Published January 21,
402 Pages 274 B/W Illustrations
This book helps students explore Fourier analysis and its related topics, helping them appreciate why it pervades many fields of mathematics, science, and engineering.
This introductory textbook was written with mathematics, science, and engineering students with a background in calculus and basic linear algebra in mind. It can be used as a textbook for undergraduate courses in Fourier analysis or applied mathematics, which cover Fourier series, orthogonal functions, Fourier and Laplace transforms, and an introduction to complex variables. These topics are tied together by the application of the spectral analysis of analog and discrete signals, and provide an introduction to the discrete Fourier transform. A number of examples and exercises are provided including implementations of Maple, MATLAB, and Python for computing series expansions and transforms.
Review of Sequences and Infinite Series. Fourier Trigonometric Series. Generalized Fourier Series and Function Spaces. Complex Analysis. Fourier and Laplace Transforms. From Analog to Discrete Signals. Signal Analysis.
ISBN 9780367497873
April 27, 2023 Forthcoming
242 Pages
This monograph, for the first time in book form, considers the large structure of metric spaces as captured by bornologies: families of subsets that contain the singletons, that are stable under finite unions, and that are stable under taking subsets of its members. The largest bornology is the power set of the space and the smallest is the bornology of its finite subsets. Between these lie (among others) the metrically bounded subsets, the relatively compact subsets, the totally bounded subsets, and the Bourbaki bounded subsets.
Classes of functions are intimately connected to various bornologies; e.g., (1) a function is locally Lipschitz if and only if its restriction to each relatively compact subset is Lipschitz; (2) a subset is Bourbaki bounded if and only if each uniformly continuous function on the space is bounded when restricted to the subset. A great deal of attention is given to the variational notions of strong uniform continuity and strong uniform convergence with respect to the members of a bornology, leading to the bornology of UC-subsets and UC-spaces. Spaces on which its uniformly continuous real-valued functions are stable under pointwise product are characterized in terms of the coincidence of the Bourbaki bounded subsets with a usually larger bornology.
Special attention is given to Lipschitz and locally Lipschitz functions. For example, uniformly dense subclasses of locally Lipschitz functions within the real-valued continuous functions, Cauchy continuous functions, and uniformly continuous functions are presented. It is shown very generally that a function between metric spaces has a particular metric property if and only if whenever it is followed in a composition by a real-valued Lipschitz function, the composition has the property. Bornological convergence of nets of closed subsets, having Attouch-Wets convergence as a prototype, is considered in detail. Topologies of uniform convergence for continuous linear operators between normed spaces is explained in terms of the bornological convergence of their graphs. Finally, the idea of a bornological extension of a topological space is presented, and all regular extensions can be so realized.
Background material. Continuous functions on metric spaces. Extension of real-valued continuous functions on subsets of a metric space. The Arens-Eells theorem and the Hausdorff extension theorem. Nets and uniformities. Some basic bornologies. Total boundedness revisited and Bourbaki boundedness. Localy Lipschitz functions. Common sets of boundedness for classes of continuous functions. Hejcmanfs theorem and its analog for totally bounded subsets. General constructions with bornologies. Properties of bornologies. Approximation by members of a bornology. Selected topological properties of the one-point extension. Bornologies of metrically bounded sets. Bornologies of totally bounded sets. Strong uniform continuity. UC-subsets. UC-spaces. Pointwise products of uniformly continuous real-valued functions. Strong uniform convergence on bornologies. Uniform convergence on totally bounded subsets. Where must each member of a class of locally Lipschitz functions be Lipschitz? Real-valued Lipschitz functions and classes of locally Lipschitz functions. Coarse maps on metrically convex spaces. Some density results. More on our 4 classes of locally Lipschitz functions. Real-valued functionals and bornologies. Uniformly paracompact subsets. Uniformly paracompact spaces and uniformly locally Lipschitz functions. Bornological convergence of nets of closed subsets. Attouch-Wets convergence. Topologies of uniform convergence on B(X,Y) and convergence of graphs. Bornological convergence and uniform convergence of distance functionals. Bornological convergence with respect to the compact bornology. When is bornological convergence topological? Uniformizability and metrizability. Ideals, bornologies and extensions. When is an extension bornological?
ISBN 9781032448336
May 12, 2023 Forthcoming
198 Pages
This book not only examines the constructions and free-probabilistic properties of semicircular elements, as defined within the text, but also considers certain Banach-space operators acting on these semicircular elements and shows how they deform (i.e., preserve-or-distort) the semicircular law induced by orthogonal projections.
1. Introduction
2. Semicircular Elements Induced by p-Adic Number Fields Qp
3. Semicircular Elements Induced by Orthogonal Projections
4. Certain Banach-Space Operators on LQ [N]
5. Discussion
ISBN 9780367507879
June 12, 2023 Forthcoming
528 Pages 486 B/W Illustrations
The interplay between graph theory and a wide variety of models and applications in mathematics, computer science, operations research, and the natural and social sciences continues to grow.
This book is geared toward the more mathematically mature student. The first two chapters provide the basic definitions and theorems of graph theory and the remaining chapters introduce a variety of topics and directions for research. @These topics draw on numerous areas of theoretical and applied mathematics, including combinatorics, probability, linear algebra, group theory, topology, operations research, and computer science. This makes the book appropriate for a first course at the graduate level or as a second course at the undergraduate level.
The authors build upon material previously published in Graph Theory and Its Applications, third edition, by the same authors. That text covers material for both an undergraduate and graduate course, while this book builds on and expands the graduate-level material.
Extensive exercises and applications.
Flexibility: appropriate for either a first course at the graduate level; or an advanced course at the undergraduate level.
Opens avenues to a variety of research areas in graph theory.
Emphasis on topological and algebraic graph theory
Foundations
Basic Definitions and Terminology
Walks and Connectivity
Subgraphs
Graph Operations
Directed Graphs
Formal Specifications for Graphs and Digraphs
Isomorphisms and Symmetry
Graph Homomorphisms and Isomorphisms
Automorphisms and Symmetry
Tests for Non-Isomorphism
Trees and Connectivity
Characterizations and Properties of Trees
Cycle, Edge-Cuts, and Spanning Trees
Graphs and Vector Spaces
Vertex- and Edge-Connectivity
Max-Min Duality and Mengerfs Theorems
Block Decompositions
Planarity and Kuratowskifs Theorem
Planar Drawings and Some Basic Surfaces
Subdivision and Homeomorphism
Extending Planar Drawings
Kuratowskifs Theorem
Algebraic Tests for Planarity
Planarity Algorithm
Crossing Numbers and Thickness
Drawing Graphs and Maps
The Topology of Low Dimensions
Higher-Order Surfaces
Mathematical Model for Drawing Graphs
Regular Maps on a Sphere
Embeddings on Higher-Order Surfaces
Geometric Drawings of Graphs
Graph Colorings
Vertex-Colorings
Local Recolorings
Map-Colorings
Edge-Colorings
Factorization
Measurement and Mappings
Distance in Graphs
Domination in Graphs
Bandwidth
Intersection Graphs
Linear Graph Mappings
Modeling Network Emulation
Analytic Graph Theory
Ramsey Theory
Extremal Graph Theory
Random Graphs
Graph Colorings and Symmetry
Automorphisms of Simple Graphs
Equivalence Classes of Colorings
Burnsidefs Lemma
Cycle-Index Polynomial of a Permutation Group
More Counting, Including Simple Groups
Polya-Burnside Enumeration
Algebraic Specification of Graphs
Cyclic Voltages
Specifying Connected Graphs
Zn-Voltage Graphs and Graph Colorings
General Voltage Graphs
Permutation Voltages
Symmetric Graphs and Parallel Architectures
Nonplanar Layouts
Representing Imbeddings by Rotations
Genus Distribution of a Graph
Voltage-Graph Specification of Graph Layouts
Non-KVL Imbedded Voltage Graphs
The Heawood Map-Coloring Problem
ISBN 9781032349169
June 6, 2023 Forthcoming
404 Pages
Geometry of Derivation with Applications is the fifth work in a longstanding series of books on combinatorial geometry (Subplane Covered Nets, Foundations of Translation Planes, Handbook of Finite Translation Planes, and Combinatorics of Spreads and Parallelisms). Like its predecessors, this book will primarily deal with connections to the theory of derivable nets and translation planes in both the finite and infinite cases. Translation planes over non-commutative skewfields have not traditionally had a significant representation in incidence geometry, and derivable nets over skewfields have only been marginally understood. Both are deeply examined in this volume, while ideas of non-commutative algebra are also described in detail, with all the necessary background given a geometric treatment.
The book builds upon over twenty years of work concerning combinatorial geometry, charted across four previous books and is suitable as a reference text for graduate students and researchers. It contains a variety of new ideas and generalizations of established work in finite affine geometry and is replete with examples and applications.
Preface
Part 1. Classical theory of derivation
Chapter 1. Coordinate methods
Translation planes and quasifibrations
Quasifields
Left quasifields
T -extension
Chapter 2. Embedding theory of derivable nets
Co-dimension 2 construction
Structure theory and contraction of embedded nets
Embedding of subplane covered nets
Transversals to derivable nets
Part 2. Classifying derivable nets over skewfields
Chapter 3. Fundamentals & background
Uniform representation for quaternion division rings
Quaternion division ring planes
Matrices and determinants over skewfields
Classifying derivable nets
Chapter 4. Classification theory over skewfields
Notation
Extension of skewfields theorem/Skewfield bimodules
Preliminary types 1, 2, 3
Standard framework
Generalized quaternions over skewfields
Matrix skewfields are generalized quaternion
Generalized (a, b)F contains (a, b)Z(F )
Brauer groups
Extending skewfields
Part 3. Types i of derivable nets
Chapter 5. The types
CONTENTS
Type 0
Double regulus type 0 derivable nets
The ambient space
Derivable nets of type 3
Order in type 3 derivable nets
Derivable nets of type 2
Fake type 2 derivable nets
Open form derivable nets of type 2
Order in type 2 derivable nets
Derivable nets of type 1
Examples of type 1 derivable nets
Carrier nets
Derivable nets in translation planes
Part 4. Flocks of a-cones
Chapter 6. Klein quadric and generalization
a-Klein quadric
Construction of general flocks
The field case
Algebraic construction for a-cones
Elation groups and flokki planes
Maximal partial spreads and a-flokki
The second cone
Baer groups for flokki Planes
q-Flokki and lifting
Collineations and isomorphisms of a-flokki planes
Part 5. Flock geometries
Chapter 7. Related geometries
Kantor's coset technique
Quasi-BLT-sets
s-Inversion & s-square
A census
Quasi-flock derivations
Herds of Hyperovals
Hyperbolic fibrations
The correspondence theorem
Flocks to cyclic planes.
Part 6. Twisted hyperbolic flocks
Chapter 8. Hyperbolic flocks and generalizations.
Algebraic theory of twisted hyperbolic flocks
Simultaneous a-Flocks & twisted hyperbolic spreads
Flocks of D-cones
j planes and twisted hyperbolic flocks
Joint theory of a-flocks
The Ka-Klein quadric
Baer theory
Quasi-flocks
The Baer forms
Algebraic and a-Klein methods
Infinite flocks of hyperbolic quadrics
Part 7. Lifting
Chapter 9. Chains & surjectivity of degree 1
1. Restricted surjectivity
2. Hughes-Kleinfield look-alikes
3. The remaining quasifibrations of dimension 2
4. Large dimension quasifibrations
5. T -copies of generalized twisted field planes
Part8. Lifting skewfields
Chapter 10. General theory
1. Matrix forms and replacement
2. The general skewfield spread
3. Generalized quaternion division rings
4. Retraction
Part 9.Bilinearity
Chapter 11. General bilinear geometries
Star flocks and rigidity
Bilinear a-flocks
Bilinear flocks of quadratic cones
Translation planes admitting SL(2, K)
Double covers
nm-Linear flocks of quadratic cones
Nests of reguli
Group replaceable translation planes
Circle geometry over K(?--)
aa-1 -nest planes
Flocks of elliptic quadrics
Klein quadric and Pappian spreads
n-Linear elliptic flocks
Tangential packings of ovoids
Part 10. Multiple replacement theorem
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