Formula counts all fullerenes with given vertices.
problem Counting all fullerenes with a given number of vertices.
method Used modular forms to derive an exact formula.
result Exact formula for the number of oriented fullerenes.
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10 results for “fullerenes”
We explore some generalizations of fullerenes F_v (simple polyhedra with v vertices and only 5- and 6-gonal faces) seen as (d-1)-dimensional simple manifolds (preferably, spherical or polytopal) with only 5- and 6-gonal 2-faces. First, finite and planar (infinite) 3-fullerenes are described. Three infinite families of …
Hyperbolic volume correlates with chemical properties of fullerenes.
problem Understanding the relationship between fullerene structure and chemical properties.
method Calculated hyperbolic volumes of fullerenes and correlated them with topological indices.
result Hyperbolic volume correlates with Wiener index and other topological indices of fullerenes.
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 and explain some new algorithms that allow their efficient enumeration. Using this we give the symmetry groups of all i-hedrites and the minimal representative for each. We also review the link of 4-hed…
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 opposite edges. A graph without a railroad is called tight. We consider the zigz…
The study examines polyhedra with hexagonal and triangular faces, focusing on their 3-regular planar graphs.
problem Analyzing polyhedra with hexagonal and triangular faces and three faces around each vertex.
method Representing polyhedra as quotients of hexagonal tilings under isometries, using signatures to describe the arrangement of rotations, and establishing a bijection between trihexes and equivalence classes of signatures.
result A bijection between trihexes and equivalence classes of signatures, allowing bounds on the number of trihexes for a given number of vertices.
Hydrogen atom confined inside an inverted-Gaussian potentialphysics.atom-ph
Hydrogen atom confined in an inverted-Gaussian potential, with detailed numerical methods and results.
problem Studying hydrogen atom in a specific potential.
method Three numerical methods: Lagrange-mesh, fourth order finite differences, and finite element method.
result Accurate numerical results for hydrogen atom energies and eigenfunctions, improving previous literature.
A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. We consider the class P of 3…
Gryffin optimizes categorical variables in materials design, leveraging expert knowledge.
problem Optimizing categorical variables in complex design choices like molecule selection.
method Bayesian optimization with smooth approximations to categorical distributions, incorporating expert knowledge.
result Gryffin accelerates discovery of promising molecules and materials, highlighting relevant correlations.
A novel symmetry in nanocarbons: pre-constant discrete principal curvature structurecond-mat.mtrl-sci
Novel symmetry found in nanocarbons' discrete principal curvature structure.
problem Identifying novel symmetries in nanocarbons' geometric structures.
method First-principles calculations and discrete geometry analysis.
result Discovery of a novel symmetry (pre-constant discrete principal curvature) in nanocarbons.