Pentagonal chains and annuli as models for designing nanostructures from cages

Abstract

Carbon is the most versatile of chemical elements in combining with itself or other elements to form chains, rings, sheets, cages, and periodic 3D structures. One of the perspective trends for creating new molecules of nanotechnological interest deals with constructs which may be formed by chemically linking of cage molecules.The growing interest to fullerene polyhedra and other molecules with pentagonal rings raises also a question about geometrically consistent in E 3 nanoarchitectures which may be obtained by aggregating many such molecules. Simple examples are chains and rings assembled from pyramidal (car)borane subunits. Adequate geometrical models of such objects are a chain and an annulus built from regular pentagons wherein any two adjacent pentagons share an edge. Among arising combinatorial problems may be both analytical and constructive enumeration of such chains and annuli drawn in plane with no two edges crossing each other. This may also employ several mathematical disciplines, such as geometry, (spectral) graph theory, semigroup theory, theory of fractals, and others.We discuss some practical approaches for solving the mentioned mathematical problem.

Enumeration of Pentahexagonal Annuli in the Plane

“…A fast algorithm for computer generation of polycyclic chains with arbitrary ring sizes was proposed in [45]. Pentahexagonal chains with up to 13 rings were counted with respect to their symmetries in [46], and some problems concerning pentagonal chains and annulus were discussed in [47]. An explicit expression of the counting formula for some hexagonal annuli of the belt-type and Möbius-type was derived in [48].…”

Extremal pentagonal chains with respect to bond incident degree indices

“…Then using this formula, we obtain the extremal values for a variety of BID indices over the certain collection of pentagonal chains with fixed number of pentagons and characterize the corresponding extremal pentagonal chains. The main motivation of considering the pentagonal chains in the present study comes from the papers [33][34][35][36] , which are related to different types of pentagonal chains.…”