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.
