Geometry of Chemical Graphs. Polycycles and Two-faced Maps
The authors present graph theory as a tool used in the natural sciences. Graphs and their embeddings arise as natural structures in chemistry and crystalography. Graphs can be viewed as models of large molecules and their (abstract) properties can describe characteristics of chemical substances. This book covers two main topics - polycycles and two-faced maps. The main notions are introduced in the first two chapters and the remaining chapters can be read almost independently. Polycycles are 2-connected plane graphs with restricted combinatorial type of interior faces and the same degree q for interior vertices, while degree is at most q for boundary vertices. The book explains a general notion of (r,q)-polycycles and describes their classification when it is possible. Two-faced maps are maps (planar, thoroidal, etc.) having at most two types of faces and the same degree of all vertices. Most classical examples of two-faced maps are fullerens (cubic graphs whose faces are pentagons and hexagons). The classification of other types of two-faced maps is presented with respect to face-regularity and symmetries.