Trivalent discrete surfaces and carbon structures / Трехвалентные дискретные поверхности и углеродные структуры

Let us start with a question here. Which figure in Fig. 1.1 is the most symmetric? Obviously, figure (c) in Fig. 1.1 has less symmetry than (a) and (b); however, it is difficult to compare symmetries of (a) and (b).
One of the basic tools for describing symmetries of crystal structures is space groups, which describe symmetry of atoms (vertices/points) in a crystal structure. For example, the symmetry of a regular hexagonal tiling of R2 is described by the group P6m, and the space group of the symmetry of a regular three-colored hexagonal tiling (see Fig. 1.2b) is P3m1. Similarly, the group P6m describes the symmetry of a regular hexagonal lattice. The groups describing the symmetry of (a) and (b) of Fig. 1.1 are P4mm and P6m, respectively, and one is never included in the other. Topological crystallography, which was pionnered by Kotani and Sunada [28–30, 54], describes symmetries of both of vertices and of edges (atomic bonds of crystal structure). Why does nature select (b) among (a)–(c) in Fig. 1.3? Note that these  lattices are created by the same graph. Topological crystallography answers thisquestion. In mathematics, a structure consisting of vertices and edges (connectivity of vertices) is called a graph, and graph theory is one of the basic tools of topological crystallography (Chap. 2). However, graphs describe only vertices and their connectivities, as in Fig. 2.2; placements of vertices and edges in Rn are not defined in the notion of graphs. Therefore, we should define placements of a given graph structure, which describes crystal structure, and should consider how to define nice placement of the graph. By defining the energy of placements of a graph, we may find a nice placement, which is called a standard realization, by using variational principles. A standard realization gives us one of the most symmetric objects among all placements of the graph (Chap. 3). In the first few sections, we discuss topological crystallography including graph theory and geometry. The most important reference of this part is Sunada’s lecture note [53]. The author discusses an introduction to topological crystallography along with it. <...>