On Cyclic Edge-Connectivity of Fullerenes

A graph is said to be cyclic $k$-edge-connected, if at least $k$ edges must be removed to disconnect it into two components, each containing a cycle. Such a set of $k$ edges is called a cyclic-$k$-edge cutset and it is called a trivial cyclic-$k$-edge cutset if at least one of the resulting two components induces a single $k$-cycle. It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclic 5-edge-connected. In this article it is shown that a fullerene $F$ containing a nontrivial cyclic-5-edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is shown that $F$ has a Hamilton cycle, and as a consequence at least $15\cdot 2^{\lfloor \frac{n}{20}\rfloor}$ perfect matchings, where $n$ is the order of $F$.Comment: 11 pages, 9 figure

The Clar Structure of Fullerenes

A fullerene is a 3-regular plane graph consisting only of pentagonal and hexagonal faces. Fullerenes are designed to model carbon molecules. The Clar number and Fries number are two parameters that are related to the stability of carbon molecules. We introduce chain decompositions, a new method to find lower bounds for the Clar and Fries numbers. In Chapter 3, we define the Clar structure for a fullerene, a less general decomposition designed to compute the Clar number for classes of fullerenes. We use these new decompositions to understand the structure of fullerenes and achieve several results. In Chapter 4, we classify and give a construction for all fullerenes on |V| vertices that attain the maximum Clar number |V|/6 - 2. In Chapter 5, we settle an open question with a counterexample: we construct an infinite family of fullerenes for which a set of faces attaining the Clar number cannot be a subset of a set of faces that attains the Fries number. We develop a method to calculate the Clar number directly for many infinite families of fullerene

Orbits of crystallographic embedding of non-crystallographic groups and applications to virology

The architecture of infinite structures with non-crystallographic symmetries can be modelled via aperiodic tilings, but a systematic construction method for finite structures with non-crystallographic symmetry at different radial levels is still lacking. This paper presents a group theoretical method for the construction of finite nested point sets with non-crystallographic symmetry. Akin to the construction of quasicrystals, a non-crystallographic group G is embedded into the point group of a higher-dimensional lattice and the chains of all G-containing subgroups are constructed. The orbits of lattice points under such subgroups are determined, and it is shown that their projection into a lower-dimensional G-invariant subspace consists of nested point sets with G-symmetry at each radial level. The number of different radial levels is bounded by the index of G in the subgroup of . In the case of icosahedral symmetry, all subgroup chains are determined explicitly and it is illustrated that these point sets in projection provide blueprints that approximate the organization of simple viral capsids, encoding information on the structural organization of capsid proteins and the genomic material collectively, based on two case studies. Contrary to the affine extensions previously introduced, these orbits endow virus architecture with an underlying finite group structure, which lends itself better to the modelling of dynamic properties than its infinite-dimensional counterpart

Dominating Sets in Plane Triangulations

In 1996, Matheson and Tarjan conjectured that any n-vertex triangulation with n sufficiently large has a dominating set of size at most n/4. We prove this for graphs of maximum degree 6.Comment: 14 pages, 6 figures; Revised lemmas 6-8, clarified arguments and fixed typos, result unchange

Limits of economy and fidelity for programmable assembly of size-controlled triply-periodic polyhedra

We propose and investigate an extension of the Caspar-Klug symmetry principles for viral capsid assembly to the programmable assembly of size-controlled triply-periodic polyhedra, discrete variants of the Primitive, Diamond, and Gyroid cubic minimal surfaces. Inspired by a recent class of programmable DNA origami colloids, we demonstrate that the economy of design in these crystalline assemblies -- in terms of the growth of the number of distinct particle species required with the increased size-scale (e.g. periodicity) -- is comparable to viral shells. We further test the role of geometric specificity in these assemblies via dynamical assembly simulations, which show that conditions for simultaneously efficient and high-fidelity assembly require an intermediate degree of flexibility of local angles and lengths in programmed assembly. Off-target misassembly occurs via incorporation of a variant of disclination defects, generalized to the case of hyperbolic crystals. The possibility of these topological defects is a direct consequence of the very same symmetry principles that underlie the economical design, exposing a basic tradeoff between design economy and fidelity of programmable, size controlled assembly.Comment: 15 pages, 5 figures, 6 supporting movies (linked), Supporting Appendi

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

Theoretical model for the realization of quantum gates using interacting endohedral fullerene molecules

We have studied a physical system composed oftwo interacting endohedral fullerene molecules for quantum computational purposes. The mutual interaction between these two molecules is determined by their spin dipolar interaction. The action of static magnetic fields on the whole system allows us to encode the qubit in the electron spin of the encased atom. We present a theoretical model which enables us to realize single-qubit and twoqubit gates through the system under consideration. Single-qubit operations can be achieved by applying to the system time-dependent microwave fields. Since the dipolar spin interaction couples the two qubit-encoding spins, two-qubit gates are naturally performed by allowing the system to evolve freely. This theoretical model is applied to two realistic architectures of two interacting endohedrals. In the first realistic system the two molecules are placed at a distance of 1.14nm. In the second design the two molecules are separated by a distance of 7nm. In the latter case the condition Δωp > > g(r) is satisfied, i.e. the difference between the precession frequencies of the two spins is much greater than the dipolar coupling strength. This allows us to adopt a simplified theoretical model for the realization of quantum gates. The realization of quantum gates for these realistic systems is provided by studying the dynamics of the system. In this extent we have solved sets of Schrodinger equations needed for reproducing the respective gates, i.e. phase-gate, 1r-gate and CNOT-gate. For each quantum gate reproduced through the realistic. system, we have estimated their reliability by calculating the related fidelity. The presented two-qubit gates are characterized by very high values of fidelity. The value of minimum fidelity related to the realization of a CNOT-gate is F=0.9995, which differs from the ideal value F=1 by of the order of 10⁻²%. We also present suggestions regarding the improvements on systems composed of endohedral fullerenes in order to enable the experimental realization of quantum gates. This would allow these systems to become reliable building blocks of a quantum computer.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Non-Euclidean Shells: A Study of Growth-Induced Fabrication and Mechanical Multi-Stability

Non-Euclidean shells are ubiquitous in both natural and man-made systems, yet fabrication at smaller length-scales (nanometer to micrometer) and the underlying mechanical behavior of these geometries is not well understood. In this dissertation, we develop a framework for improvements in fabrication and control over deformation pathways of non-Euclidean elastic shells. For programming a target non-Euclidean geometry, we study the non-uniform growth induced buckling in flat sheets. To deepen our understanding of this powerful mechanism, we present an experimental verification of its mathematical equivalence with a mechanism involving topological defects. We establish a framework for correlating topological defect-induced buckling, realized through a simple cut-and-glue construction, with growth induced buckling realized through non-uniform growth of patterned 2D hydrogel sheets. Validating the obtained mathematical results, we demonstrate fabrication of a cylindrical and conical dipole geometry through both mechanisms under consideration. In addition, upon applying a similar treatment on tetrahedron geometry we realize the limitations of growth-induced buckling mechanism for programming 3D geometry, and find that optimizing for pattern resolution and swelling range can lead to a higher fidelity in target geometries. Next, we turn towards the interplay between geometry and mechanics in non-Euclidean shells, to harness multi-stability between different geometric configurations. Under this theme, we study deformation of arbitrarily curved surfaces along pre-defined creases (curves with local weakening) finding that the continuity of this deformation can be predicted through a simple consideration of the curvature of the crease. We establish that simple geometric control over the crease curvature can be used to program on-demand snap-through instabilities between bi-stable states of developable, elliptic and hyperbolic surfaces. Using experiments, FEA and theoretical analysis of bending and stretching energies involved while indenting a hemispherical shell, we establish the geometric phase space in which the isometric state of a creased sphere is stable. Finally, we extend this notion by considering the ability to program multiple stable states in a tiled conical geometry, commonly found in bendable drinking straws (bendy straws) and other similar structures. These corrugated structures exhibit stability in axial (resulting in change in length), non-axial (change in ‘bent’ angle) and azimuthal direction (change in azimuthal angle) imparting a desirable high-degree of freedom in possible configurations. By analyzing the stability behavior of elastic double conical frusta shells, we study the necessary conditions for programming multi-stability, and find that axial stability depends on geometrical parameters. Interestingly, we conclude that a stable non-axial state requires a combination of appropriate geometry and presence of a pre-stress in azimuthal direction. The controlled multi-stability opens pathways toward a truly re-configurable shape programmable system, useful for manipulators and actuators in soft-robotic