Structural features of cubic “polyhedra” up to 24 vertices with four-, five- and six-sided faces

Abstract

We present a complete list of three-dimensional structures corresponding to all cubic polyhedral graphs with the number of vertices up to 24, which have only square, pentagonal and hexagonal cycles. It can be considered as an Atlas of possible small- and middle-sized cages of three-dimensional 4-coordinated frameworks. Adhering to graph-theoretical terminology, we use the term “polyhedron”. But, strictly speaking, most of the constructed three-dimensional figures are not polyhedra because their “faces” are not planar. In addition, many polyhedra are significantly different in shape from conventional fullerenes. The spiral codes are used as the main topological identifier of the polyhedra. Three-dimensional coordinates of polyhedra are calculated by minimizing the sum of the squared deviations of the edge lengths and distances between the second neighbors from the idealized values. It is concluded that the minimum value of this sum can be considered as a new supplementary identifier of polyhedra and as a measure of deviation of the polyhedral structure from the set of true face-regular polyhedra. An algorithm for constructing polyhedral structures with edges of the same length is proposed. Also, we discuss the geometric properties of the constructed polyhedra and their relation to the stability of molecular systems.

Keywords:

• Polyhedra
• graphs
• spiral codes
• polyhedron construction
• face regularity

Additional information

Funding

The present work was supported by the Basic Research Program of RAS No. IX.135.2.3.