Polytopes and Graphs

An introduction to convex polytopes and their graphs, including both background material and cutting-edge research.

Guillermo Pineda Villavicencio (Author)

9781009257817, Cambridge University Press

Hardback, published 21 March 2024

480 pages
23.5 x 15.8 x 3 cm, 0.84 kg

'… Villavicencio has produced a most welcome addition to the mathematical literature on polytopes. It is accessible and thorough, and it underscores developments from the past two decades: in other words, this book fills a gap within the existing set of texts. The book points the interested reader in numerous directions for further investigation and provides challenging exercises. … I highly recommend it to anyone wishing to explore the world of polytopes, especially students.' Robert Davis, Notices of the American Mathematical Society

This book introduces convex polytopes and their graphs, alongside the results and methodologies required to study them. It guides the reader from the basics to current research, presenting many open problems to facilitate the transition. The book includes results not previously found in other books, such as: the edge connectivity and linkedness of graphs of polytopes; the characterisation of their cycle space; the Minkowski decomposition of polytopes from the perspective of geometric graphs; Lei Xue's recent lower bound theorem on the number of faces of polytopes with a small number of vertices; and Gil Kalai's rigidity proof of the lower bound theorem for simplicial polytopes. This accessible introduction covers prerequisites from linear algebra, graph theory, and polytope theory. Each chapter concludes with exercises of varying difficulty, designed to help the reader engage with new concepts. These features make the book ideal for students and researchers new to the field.

Preface
1. Introduction
2. Polytopes
3. Polytopal graphs
4. Connectivity
5. Reconstruction
6. Decomposition
7. Diameter
8. Faces
A. Open problems
B. Topology
C. Graphs
References
Glossary
Index.

Subject Areas: Discrete mathematics [PBD]