Polytopes and Graphs Cambridge Studies in Advanced Mathematics Series

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.

Guillermo Pineda Villavicencio is Senior Lecturer in Computer Science and Mathematics at Deakin University (Australia), an honorary senior research fellow at Federation University Australia, and a fellow of AdvancedHE. 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 was a member of its Council from 2018 to 2022. He is also a life member of the Combinatorial Mathematics Society of Australasia.