Trivalent Discrete Surfaces and Carbon Structures (SpringerBriefs in the Mathematics of Materials, 5)

✍ Scribed by Hisashi Naito

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 113
• Category: Library

✦ Synopsis

This book discusses discrete geometric analysis, especially topological crystallography and discrete surface theory for trivalent discrete surfaces. Topological crystallography, based on graph theory, provides the most symmetric structure among given combinatorial structures by using the variational principle, and it can reproduce crystal structures existing in nature.
In this regard, the topological crystallography founded by Kotani and Sunada is explained by using many examples. Carbon structures such as fullerenes are considered as trivalent discrete surfaces from the viewpoint of discrete geometric analysis. Discrete surface theories usually have been considered discretization of smooth surfaces. Here, consideration is given to discrete surfaces modeled by crystal/molecular structures, which are essentially discrete objects.

✦ Table of Contents

Preface
Contents
1 Overview of This Monograph
2 Graph Theory
2.1 Graph
2.2 Covering Graph
2.3 Graph Laplacian
3 Topological Crystals
3.1 Topological Crystals and Their Standard Realizations
3.2 Construction of Standard Realizations
3.2.1 Explicit Calculations in Cases of Maximal Abelian Coverings
3.2.2 Explicit Calculations for Generic Cases
3.3 Carbon Structures and Standard Realizations
4 Negatively Curved Carbon Structures
4.1 Carbon Structures as Discrete Surfaces
4.2 Constructions of Negatively Curved Carbon Structures …
5 Trivalent Discrete Surfaces
5.1 Curvatures of Trivalent Discrete Surfaces
5.2 Examples of Trivalent Discrete Surfaces and Their Curvatures
5.3 Construction of Trivalent Minimal Discrete Surfaces
6 Subdivisions of Trivalent Discrete Surfaces
6.1 Goldberg–Coxeter Constructions
6.2 Subdivisions of Trivalent Discrete Surfaces
6.3 Another Method of Subdivision of Trivalent Discrete Surfaces
7 Miscellaneous Topics
7.1 Carbon Nanotubes from Geometric Viewpoints
7.2 Material Properties and Discrete Geometry
7.3 Graph Spectra and Electronic Properties of Carbon Structures
7.3.1 The Hückel Method
7.3.2 Tight-Binding Approximations
7.3.3 The Density Function Theory
7.4 Eigenvalues of Subdivisions of Trivalent Discrete Surfaces
7.5 Graph Algorithms and Miscellaneous Computation Methods
7.5.1 Graph Algorithms
7.5.2 Numerical Methods for Matrices
Appendix References
Index