Topological Theory of Graphs 1st Edition by Yanpei Liu ISBN 311047669X 9783110476699

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ISBN 10: 311047669X

ISBN 13: 9783110476699 

Author: Yanpei Liu 

This book presents a topological approach to combinatorial configurations, in particular graphs, by introducing a new pair of homology and cohomology via polyhedra. On this basis, a number of problems are solved using a new approach, such as the embeddability of a graph on a surface (orientable and nonorientable) with given genus, the Gauss crossing conjecture, the graphicness and cographicness of a matroid, and so forth. Notably, the specific case of embeddability on a surface of genus zero leads to a number of corollaries, including the theorems of Lefschetz (on double coverings), of MacLane (on cycle bases), and of Whitney (on duality) for planarity. Relevant problems include the Jordan axiom in polyhedral forms, efficient methods for extremality and for recognizing a variety of embeddings (including rectilinear layouts in VLSI), and pan-polynomials, including those of Jones, Kauffman (on knots), and Tutte (on graphs), among others.

Table of contents:

1. Preliminaries

2. Polyhedra

3. Surfaces

4. Homology on Polyhedra

5. Polyhedra on the Sphere

6. Automorphisms of a Polyhedron

7. Gauss Crossing Sequences

8. Cohomology on Graphs

9. Embeddability on Surfaces

10. Embeddings on Sphere

11. Orthogonality on Surfaces

12. Net Embeddings

13. Extremality on Surfaces

14. Matroidal Graphicness

15. Knot Polynomials

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Tags: Yanpei Liu, Topological Theory, Graphs