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.
- 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 Menger’s Theorems
- Block Decompositions
- Planarity and Kuratowski’s Theorem
- Planar Drawings and Some Basic Surfaces
- Subdivision and Homeomorphism
- Extending Planar Drawings
- Kuratowski’s 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
- Burnside’s 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
Biography
Mark Anderson is a professor of mathematics and computer science at Rollins College. His research interests in graph theory center on the topological or algebraic side.
Jonathan L. Gross
is a professor of computer science at Columbia University. His research interests include topology and graph theory.Jay Yellen is a professor of mathematics at Rollins College. His current areas of research include graph theory, combinatorics, and algorithms.
