Algebraic Elements of Graphs

Yanpei Liu, University of Science and Technology China Press

• Publisher: De Gruyter
• Year: 2017
• Print ISBN: 9783110480733
• eBook ISBN: 9783110480757
• Edition: 1
• Topic: Matemáticas
• Subtopic: Álgebra

This book studies algebraic representations of graphs in order to investigate combinatorial structures via local symmetries. Topological, combinatorial and algebraic classifications are distinguished by invariants of polynomial type and algorithms are designed to determine all such classifications with complexity analysis. Being a summary of the author's original work on graph embeddings, this book is an essential reference for researchers in graph theory.

Contents
Abstract Graphs
Abstract Maps
Duality
Orientability
Orientable Maps
Nonorientable Maps
Isomorphisms of Maps
Asymmetrization
Asymmetrized Petal Bundles
Asymmetrized Maps
Maps within Symmetry
Genus Polynomials
Census with Partitions
Equations with Partitions
Upper Maps of a Graph
Genera of a Graph
Isogemial Graphs
Surface Embeddability

1Abstract Graphs

–A graph is considered to be a partition on the union of sets obtained from each element of a given set the binary group B = {0, 1} sticks on.

–A surface, i.e., a compact two-manifold without boundary in topology, is seen as a polygon of even edges identified pairwise.

–An embedding of a graph on a surface is represented by a joint tree of the graph. A joint tree of a graph consists of a plane extended tree with labeled cotree semiedges. Two semiedges of a cotree edge have the same label as the cotree edge with a binary index. An extended tree is compounded of a spanning tree with cotree semiedges as articulate edges.

–Combinatorial properties of an embedding in abstraction are particularly discussed for the formal definition of a map.

1.1Graphs and networks

Let X be a finite set. For any x ∈ X, the binary group B = {0, 1} sticks on (seen as a type of group action!) x to obtain Bx = {x(0), x(1)}. x(0) and x(1) are called the ends of x, or Bx. If Bx is seen as an ordered set 〈x(0), x(1)〉, then x(0) and x(1) are, respectively, initial and terminal ends of x. Let

$X={\displaystyle \sum _{x\in X}Bx,}\text{...}$

Table of contents

• Cover
• Title Page
• Copyright
• Contents
• 1 Abstract Graphs
• 2 Abstract Maps
• 3 Duality
• 4 Orientability
• 5 Orientable Maps
• 6 Nonorientable Maps
• 7 Isomorphisms of Maps
• 8 Asymmetrization
• 9 Asymmetrized Petal Bundles
• 10 Asymmetrized Maps
• 11 Maps within Symmetry
• 12 Genus Polynomials
• 13 Census with Partitions
• 14 Equations with Partitions
• 15 Upper Maps of a Graph
• 16 Genera of a Graph
• 17 Isogemial Graphs
• 18 Surface Embeddability
• Appendix 1: Concepts of Polyhedra, Surfaces, Embeddings and Maps
• Appendix 2: Table of Genus Polynomials for Embeddings and Maps of Small Size
• Appendix 3: Atlas of Rooted and Unrooted Maps for Small Graphs
• Bibliography
• Author Index
• Subject Index