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A counting polynomial C(G,x) is a sequence description of a topological property so that the exponents express the extent of its partitions while the coefficients are related to the occurrence of these partitions. Basic definitions and properties of the Omega polynomial Ω(G,x) and Sadhana polynomial Sd(G,x) are presented. These polynomials for some infinite classes of fullerenes and nanotubes are...
A topological index for a molecular graph is a number correlated some physic-chemical properties of the molecule under consideration. The Wiener index is the first reported distance based topological index applicable in chemistry. It is defined as half sum of the distances between all the pairs of vertices in a molecular graph. In this survey article a brief account on the development of the Wiener...
It was shown that the covering of C60 “Buckminsterfullerene” is basically sumanenic, with the empty π-electron faces being only pentagons. Four series of cages, tessellated by sumanenic patterns S[r] = [r:(5,6)r/2], were generated by sequences of map operations, and their topology described. Among these cages, which all show all_R[5] 2-factors, those belonging to the series designed...
We present a general method which enables a possible classification of fullerenes by means of local topological invariants. In this study fullerenes are considered as bifaced simple (trivalent) polyhedra. The method proposed is based on the combinatorial analysis of the first neighbor environments (coronas) of vertices and/or edges of bifaced simple polyhedra. For this purpose, we used the so-called...
In this chapter, we give a GAP program for computing the Schultz index, Modified Schultz index, Wiener index, hyper Wiener index, Balaban index and Zagreb indices for any graph and by this algorithm we compute the Schultz polynomial and Schultz index of C60 and C80 fullerene by GAP program. Also we compute the Wiener index, hyper Wiener index and Wiener polynomial of C80 fullerene by this program...
An i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do a short survey of their known properties (Deza et al. Proceedings of ICM Satellite Conference On Algebra and Combinatorics, 2003b; Deza et al. J Math Res Expo 22:49,2002; Deza and Shtogrin, Polyhedra in Science and Art 11:27, 2003a) and explain some new algorithms that allow their efficient enumeration. Using this we give...
As shown by our group (Dodziuk and Nowinksi, 1998; Dodziuk et al., manuscript in preparation, 2011) endohedral fullerene complexes are objects of nontrivial topology. An insertion of a guest (atom, ion, or molecule(s)) inside the fullerene cage usually changes properties of both host and guest. Due to their size and complexity, predicting properties of the complexes is a difficult task. For instance,...
Buckminsterfullerene has 12500 Kekulé structures grouped in 158 isomorphic classes. In this paper we reproduce the results of paper (Vukičević et al. Croatica Chemica Acta 78: 223, 2005) with some extensions. Namely, for each Kekulé structure we provide: number of structures isomorphic to it, the average number of π-electrons that belong to hexagon, the average number of π-electrons that belong to...
The recently developed idea of analyzing complex networks in terms of node displacement due to vibration (Estrada and Hatano, Chem Phys Lett 486:166–170, 2010a) is applied to fullerenes. The fact that the ramafullerenes (fullerenes of Ramanujan graphs) are limited to fullerenes with relatively small number of C atoms is explained from the point of view of the node displacement. The node displacement...
The Cycle Theorem was introduced under that name by Kirby, Klein, Mallion, Pollak and Sachs in 2004. It provides a formula for calculating how many spanning trees a graph has (its complexity) but, for some graphs, even quite small ones, the calculation is a laborious and error-prone process to do by hand. A simple algorithm for computer application is developed, which uses the Cycle Theorem to calculate...
This article presents a general topological computational method for the exact determination of the number of 13C–NMR resonance peaks and their relative intensities and applies it to the case of C66 isomers. Heuristically, the joint usage of Wiener-based topological invariants of dual and direct molecular graphs provides quick simulations of its 13C–NMR spectrum of a given fullerene. Topological tools...
About 10 years ago the synthesis of random carbon schwarzites by supersonic cluster beam deposition has endowed the rich sp2 carbon family with its three-dimensional member. Its reluctance to grow as a three-periodic minimal surface according to topological and physical predictions still prevents schwarzites from being a hot topic, although spongy carbon is already having countless applications. Understanding...
By using the technique of topological symmetry the characteristic polynomials of highly symmetrical π-electron carbon networks of regular and semi-regular polyhedra were factored out, and their stability on MO basis was analyzed and discussed. Besides the soccer ball-shaped C60 fullerene, a π-electron system of truncated octahedron-shaped C24 was suggested to be a candidate for a stable spherical...
Estrada Index, EE(G), defined as the sum of exponentials of the eigenvalues of the adjacency matrix of graph G, is calculated for sets of general cubic polyhedra, and for general and isolated-pentagon fullerenes. Amongst small cubic polyhedra, the “near-fullerenes” and fullerenes minimise EE. Amongst fullerenes, the isolated-pentagon fullerenes minimise EE. The preference for fullerenes over non-fullerenes...
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