Patches are 2-connected plane graphs where all faces have the same size except one outer face and possibly some defective faces, and all vertices have the same degree except the vertices in the outer face and possibly some defective vertices. The term boundary of a patch refers to the sequence of vertex degrees in the outer face. Various applications of patches can be found in theoretical chemistry, especially in the context of fullerenes, that are spherically shaped carbon molecules where each atom is bonded to three others in a way that only hexagonal and pentagonal rings occur.

One focus of the thesis is the investigation of the number of faces in a patch with respect to its boundary. Previous results are generalized by showing that the number of faces is uniquely determined by the boundary in case a fixed subpatch containing all disorder is given. Further results concern the minimal boundary length of patches relative to their numbers of faces: Formulas for minimal boundary lengths of triangle-patches with defective vertices are developed and patches with such boundary lengths are constructed. Additionally, the case is considered where a subpatch containing the defective vertices is given.

Similar methods are applied in order to prove a result for nanocones which is of interest in chemistry: It is shown that given a hexagon-cone, that is an infinite trivalent plane graph with up to five pentagons and all other faces hexagons, there exists always a subpatch containing the pentagons which has a certain boundary that can uniquely be described by two parameters. This outcome enables a classification of these cones.

Finally, some of the developed results and techniques are applied for the determination of the expander constant of fullerenes. This constant measures the connectivity of a graph and is particularly interesting in the case of fullerenes for mathematical and chemical reasons. The expander constants of a class of symmetrical fullerenes including the Buckminster fullerene are determined. As a second approach, an algorithm is presented that is used to compute the expander constants of fullerenes. Results include a complete list of the maximal expander constants among fullerenes of the same size for all numbers of vertices until 140.