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 in fullerene polyhedra and other molecules with pentagonal rings raises also a question about geometrically consistent in E3E3 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.
The final publication is available at Springer via http://dx.doi.org/10.1007/s10910-015-0584-5
A periodic linear chain consists of a weighted 2n -path where new edges have been added following a certain periodicity. In this paper, we obtain the effective resistance and the Kirchhoff index of a periodic linear chain as non trivial functions of the corresponding expressions for the path. We compute the expression of the Kirchhoff index of any homogeneous and periodic linear chain which generalizes the previously known results for ladder-like and hexagonal chains, that correspond to periods one and two respectively
Vives, D.; Armero, J.; Martí, A.; Ramirez de La Piscina, L.; Casademunt, J.; Sancho, J M; Sagués, F. Journal of mathematical chemistry Vol. 23, num. 1,2, p. 239-260 DOI: 10.1023/A:1019152720718 Data de publicació: 1998-05 Article en revista