Related papers: Some open mathematical problems on fullerenes
We predict the stability of a new extended two-dimensional hydrocarbon on the basis of first-principles total energy calculations. The compound that we call graphane is a fully saturated hydrocarbon derived from a single graphene sheet with…
A general treatment of the spectral problem of quantum graphs and tight-binding models in finite Hilbert spaces is given. The direct spectral problem and the inverse spectral problem are written in terms of simple algebraic equations…
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies…
Graph invariants provide a powerful analytical tool for investigation of abstract structures of graphs. They, combined in convenient relations, carry global and general information about a graph and its various substructures such as cycle…
At half filling, the electronic structure of graphene can be modelled by a pair of free two-dimensional Dirac fermions. We explicitly demonstrate that in the presence of a geometrically induced gauge field, an everywhere-real Kekule…
We study a versatile structurally favorable periodic $sp^2$-bonded carbon atomic planar sheet with $C_{4v}$ symmetry by means of the first-principles calculations. This carbon allotrope is composed of carbon octagons and squares with two…
Odd electrons of benzenoid units and correlation of these electrons having different spins are the main concepts of the molecular theory of graphene. In contrast to the theory of aromaticity, the molecular theory is based on the fact that…
We consider the following question, motivated by the enumeration of fullerenes. A fullerene patch is a 2-connected plane graph G in which inner faces have length 5 or 6, non-boundary vertices have degree 3, and boundary vertices have degree…
A few years ago, by means of first-principles calculations, Enyashin et al.(2011) proposed several novel monolayers of carbon containing rings other than hexagons. One of those monolayers containing tetragons and octagons was investigated…
We consider facet-Hamiltonian cycles of polytopes, defined as cycles in their skeleton such that every facet is visited exactly once. These cycles can be understood as optimal watchman routes that guard the facets of a polytope. We consider…
We describe invariant principal and Cartan connections on homogeneous principal bundles and show how to calculate the curvature and the holonomy; in the case of an invariant Cartan connection we give a formula for the infinitesimal…
The paper is an attempt to apply the theory of dessins d'enfants to the theory of fullerenes. The classical results concerning the calculation of the dodecahedron Belyi function are presented and then applied to the calculation of the Belyi…
Molecules intercalating two-dimensional (2D) materials form complex structures that have been mostly characterized by spatially averaged techniques. Here we use aberration-corrected scanning transmission electron microscopy and…
A graph drawing in the plane is called an almost embedding if the images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. Almost embeddings (more precisely, their higher-dimensional analogues) naturally appear in…
A series of carbon allotropes with novel optoelectronic and rich topological properties is predicted by systematic first-principles calculations. These fascinating carbon allotropes can be derived by inserting acetylenic linkages…
Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory,…
A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on…
In this letter we study the electronic structures and optical properties of partially and fully fluorinated graphene by a combination of abinitio G0W0 calculations and large-scale multi-orbital tight-binding simulations. We find that for…
A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible…
We describe a new construction algorithm for the recursive generation of all non-isomorphic IPR fullerenes. Unlike previous algorithms, the new algorithm stays entirely within the class of IPR fullerenes, that is: every IPR fullerene is…