Polytopes and Graphs

Guillermo Pineda Villavicencio (Deakin University, Victoria)

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English

Cambridge University Press
21 March 2024

Discrete mathematics; Calculus & mathematical analysis; Geometry; Topology; Combinatorics & graph theory

Series: Cambridge Studies in Advanced Mathematics

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.

By: Guillermo Pineda Villavicencio (Deakin University Victoria)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Dimensions: Height: 235mm, Width: 158mm, Spine: 30mm
Weight: 840g
ISBN: 9781009257817
ISBN 10: 1009257811
Series: Cambridge Studies in Advanced Mathematics
Pages: 480
Publication Date: 21 March 2024
Audience: College/higher education , A / AS level , Further / Higher Education
Format: Hardback
Publisher's Status: Active

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.

Guillermo Pineda Villavicencio is an Associate Professor in Computer Science and Mathematics at Deakin University, Australia, and a Fellow of AdvanceHE. He conducts research on graph theory and discrete geometry, the construction and analysis of large networks, and applications of mathematics to health informatics. He is an Accredited Member of the Australian Mathematical Society and served on its Council from 2018 to 2022. He is also a Life Member of the Combinatorial Mathematics Society of Australasia.

Reviews for Polytopes and Graphs

'This is a welcome monograph on combinatorial aspects of polytope theory, with a focus on graphs. While the book is written to be accessible to non-experts, it does capture a lot of the research on that subject from the last 30 years.' Michael Joswig, Technische Universität Berlin and Max-Planck-Institute for Mathematics in the Sciences, Leipzig 'Polytopes are geometric objects that have fascinated mathematicians since antiquity. Graphs are combinatorial objects that are ubiquitous throughout pure and applied mathematics. Ever since Steinistz's famous theorem asserting that planar 3-connected graphs are precisely graphs of three-dimensional polytopes, the match between polytopes and graphs was engraved in heaven. Guillermo Pineda Villavicencio, a notable researcher exploring this connection, has authored a remarkable book on the subject. This book is well-suited for advanced undergraduates as well as professional mathematicians. It commences with a comprehensive introduction to polytopes and proceeds to explore the properties of polytopal graphs. Additionally, it delves into advanced topics, such as connectivity, diameter, reconstruction, decomposition, and more. I highly recommend this book.' Gil Kalai, Hebrew University of Jerusalem and Reichman University