Symmetry in Graphs

This book is the first full-length treatment of symmetry in graphs, a branch of algebraic graph theory that studies highly symmetric graphs and combinatorial structures using group-theoretic techniques. It assumes a first course in graph theory and group theory and covers basic material, major problems, and active research themes, with over 450 exercises.

Format: Hardback
Length: 450 pages
Publication date: 12 May 2022
Publisher: Cambridge University Press

This is the first comprehensive book on the major theme of symmetry in graphs, a branch of algebraic graph theory. It is concerned with the study of highly symmetric graphs, particularly vertex-transitive graphs, and other combinatorial structures, primarily through group-theoretic techniques. In practice, these investigations shed new light on permutation groups and related algebraic structures. The book assumes a first course in graph theory and group theory but no specialized knowledge of the theory of permutation groups or vertex-transitive graphs. It begins with the basic material before introducing the field's major problems and most active research themes to motivate the detailed discussion of individual topics that follows. Featuring many examples and over 450 exercises, it is an essential introduction to the field for graduate students and a valuable addition to any algebraic graph theorist's bookshelf.

Introduction:
Symmetry is a fundamental concept in mathematics and physics, and it has played a crucial role in the development of various fields, including graph theory. Graphs are a fundamental mathematical structure that represents relationships between objects or entities. Symmetry in graphs refers to the property of a graph that remains unchanged when it is rotated, reflected, or translated.

The Study of Symmetry in Graphs:
The study of symmetry in graphs has been an active area of research for many years. Graphs can be symmetric in various ways, including vertex-symmetric, edge-symmetric, and graph-symmetric. Vertex-symmetric graphs are those in which the vertices are identical, while edge-symmetric graphs are those in which the edges are identical. Graph-symmetric graphs are those in which the graph is the same regardless of the orientation or position of the vertices.

Vertex-Transitive Graphs:
Vertex-transitive graphs are a special class of symmetric graphs that have been studied extensively. A graph is vertex-transitive if, for any two vertices, there exists a path that connects them. Vertex-transitive graphs are important in various applications, including graph theory, computer science, and physics.

Group-Theoretic Techniques:
Group-theoretic techniques have been widely used to study symmetry in graphs. Group theory is a branch of mathematics that studies the properties of groups, which are collections of objects that can be operated on using mathematical operations. Group-theoretic techniques allow researchers to analyze the structure and properties of symmetric graphs using group-theoretic concepts such as subgroups, homomorphisms, and automorphisms.

Applications of Symmetry in Graphs:
Symmetry in graphs has numerous applications in various fields. In graph theory, symmetric graphs have been used to study the structure and properties of graphs, including the enumeration of graphs, the study of graph isomorphism, and the study of graph algorithms. In computer science, symmetric graphs have been used to study the structure and properties of networks, including the design of routing algorithms and the study of network security. In physics, symmetric graphs have been used to study the structure and properties of physical systems, including the study of quantum mechanics and the theory of relativity.

Conclusion:
Symmetry in graphs is a fascinating and important area of research that has numerous applications in various fields. The study of symmetry in graphs has led to the development of new techniques and concepts in graph theory, computer science, and physics. Symmetric graphs are important in understanding the structure and properties of graphs and have numerous applications in various fields. As research in this area continues, we can expect to see new discoveries and applications that will further our understanding of symmetry in graphs and its applications.

Weight: 869g
ISBN-13: 9781108429061