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Topics in Graph Theory

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The interplay continues to grow between graph theory and a wide variety of models and applications in mathematics, computer science, operations research, and the natural and social sciences.

Topics in Graph Theory is geared toward the more mathematically mature student. The first three 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.

Features

    • 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.

    Author(s): Jonathan L Gross, Jay Yellen, Mark Anderson
    Series: Discrete Mathematics and Its Applications
    Publisher: CRC Press/Chapman & Hall
    Year: 2023

    Language: English
    Pages: 525
    City: Boca Raton

    Cover
    Half Title
    Series Page
    Title Page
    Copyright Page
    CONTENTS
    PREFACE
    1. FOUNDATIONS
    1.1. Basic Definitions and Terminology
    1.2. Walks and Connectivity
    1.3. Subgraphs
    1.4. Graph Operations
    1.5. Directed Graphs
    1.6. Formal Specifications for Graphs and Digraphs
    Glossary
    2. ISOMORPHISMS AND SYMMETRY
    2.1. Graph Homomorphisms and Isomorphisms
    2.2. Automorphisms and Symmetry
    2.3. Tests for Non-Isomorphism
    Glossary
    3. TREES AND CONNECTIVITY
    3.1. Characterizations and Properties of Trees
    3.2. Cycles, Edge-Cuts, and Spanning Trees
    3.3. Graphs and Vector Spaces
    3.4. Vertex- and Edge-Connectivity
    3.5. Max-Min Duality and Menger's Theorems
    3.6. Block Decompositions
    Glossary
    4. PLANARITY AND KURATOWSKI'S THEOREM
    4.1. Planar Drawings and Some Basic Surfaces
    4.2. Subdivision and Homeomorphism
    4.3. Extending Planar Drawings
    4.4. Kuratowski's Theorem
    4.5. Algebraic Tests for Planarity
    4.6. Planarity Algorithm
    4.7. Crossing Numbers and Thickness
    Glossary
    5. DRAWING GRAPHS AND MAPS
    5.1. Topology of Low Dimensions
    5.2. Higher-Order Surfaces
    5.3. Mathematical Model for Drawing Graphs
    5.4. Regular Maps on a Sphere
    5.5. Imbeddings on Higher-Order Surfaces
    5.6. Geometric Drawings of Graphs
    Glossary
    6. GRAPH COLORINGS
    6.1. Vertex-Colorings
    6.2. Local Recolorings
    6.3. Map-Colorings
    6.4. Edge-Colorings
    6.5. Factorization
    Glossary
    7. MEASUREMENT AND MAPPINGS
    7.1. Distance in Graphs
    7.2. Domination in Graphs
    7.3. Bandwidth
    7.4. Intersection Graphs
    7.5. Graph Homomorphisms
    7.6. Modeling Network Emulation
    Glossary
    8. ANALYTIC GRAPH THEORY
    8.1. Ramsey Theory
    8.2. Extremal Graph Theory
    8.3. Random Graphs
    Glossary
    9. GRAPH COLORINGS AND SYMMETRY
    9.1. Automorphisms of Simple Graphs
    9.2. Equivalence Classes of Colorings
    9.3. Burnside's Lemma
    9.4. Cycle-Index Polynomial of a Permutation Group
    9.5. More Counting, Including Simple Graphs
    9.6. Polya-Burnside Enumeration
    Glossary
    10. ALGEBRAIC SPECIFICATION OF GRAPHS
    10.1. Cyclic Voltages
    10.2. Specifying Connected Graphs
    10.3. Zn-Voltage Graphs and Graph Colorings
    10.4. General Voltage Graphs
    10.5. Permutation Voltages
    10.6. Symmetric Graphs and Parallel Architectures
    Glossary
    11. NON-PLANAR LAYOUTS
    11.1. Representing Imbeddings by Rotations
    11.2. Genus Distribution of a Graph
    11.3. Voltage-Graph Specification of Graph Layouts
    11.4. Non-KVL-Imbedded Voltage Graphs
    11.5. Heawood Map-Coloring Problem
    Glossary
    A: APPENDIX
    A.1. Logic Fundamentals
    A.2. Relations and Functions
    A.3. Some Basic Combinatorics
    A.4. Algebraic Structures
    A.5. Algorithmic Complexity
    A.6. Supplementary Reading
    BIBLIOGRAPHY
    A.1. General Reading
    A.2. References
    INDEX