Goals

Press Release

Chemistry is the study of matter and its transformations, and is an experimental science, but it has an underlying mathematical structure. In this workshop, specialists from chemistry, mathematics and computer science, from research departments across the world, have come together to work on ways of improving the predictive power of chemical theories. The main target is the prediction of structures and properties of carbon nanostructures.  Carbon is the most versatile of chemical elements in combining with itself to form chains, rings, sheets, cages and periodic 3D structures. New forms of carbon, discovered in the past two decades, have potential applications in molecular electronics, chemistry and materials science. Exploitation of these opportunities will depend on collaborations of scientists from many disciplines.

Objectives

1. The first objective is to identify new classes of graphs that will be of interest as representations of chemical structure, and to design programs that generate the members of these classes. Progress in experiment means that new classes of graphs are continually being added to the list of mathematical objects of interest in chemistry. In the field of polycyclic aromatic hydrocarbons, for example, chemistry is moving beyond simple benzenoids to helicenes (`spiral staircase' benzenoids), corononoids (benzenoids with holes), benzenoids incorporating pentagonal rings. Non-classical fullerenes, azulenoid nanotori and pentaheptite graphene sheets all require systematic incorporation of pentagons and heptagons. Carbomers (decorated versions of carbon networks with degree-2 vertices decorating the edges) have been synthesised. Heterocyclic systems where some or all carbon atoms are replaced by other elements correspond to graph colorings. The objective is to identify the generalisations that are most likely to be chemically fruitful and devise efficient constructions.

2. The second objective is to identify those chemical graph invariants that contain independent information and start to correlate them with available theoretical and experimental data. Whilst it seems generally agreed that graph invariants can be used to gain qualitative understanding of large classes of molecules, there is little agreement on the invariants to be used. Adjacency eigenvalues are useful for understanding electronic structure of conjugated systems, but systems such as fullerenes have significant `steric strain' which must be captured by other parameters. Conjecturing engines have a role to play here in helping to find relationships between literature invariants and suggest modifications, and sharpen chemical intuition. Experts on the chemical invariants and researchers in conjecture-making have indicated intention to attend.

3. The third objective is to design generation algorithms for carbon structures with specified properties. The usual - and inefficient - approach is to generate all structures in a class and filter the ones with desired properties. Chemists often want extremal graphs of some kind - benzenoids with maximum number of perfect matchings, fullerenes with maximum `graph energy', singular graphs with maximum number of zero entries in the zero-eigenvalue eigenvector. Replacing the brute-force approach of post-generation structure filtering with efficient properties-driven generation will allow the generation and investigation of larger numbers of desired structures.

4. The fourth objective is to develop a unified mathematical (graph-theoretical) of molecular currents in polycyclic molecules. New theories of currents induced in polycyclic molecules by magnetic fields (ring currents) based on conjugated circuits have appeared recently in the  chemical literature, and can be put on a single combinatorial footing by considering perfect matchings. A new simple model of ballistic currents through molecules has been formulated in terms of characteristic polynomials of vertex deleted subgraphs. It is physically plausible that the two types of conduction in conjugated systems are related. Amongst intending participants are the inventors of the main simplified models for both types of conduction (Randic, Ernzerhof) and chemists, mathematicians and computer scientists who have worked on one or both (Fowler, Sciriha, Pisanski, Myrvold).

5. The fifth objective is to map the defects in fullerenes. Chemical, geometric and energetic properties of chemical systems are often governed by the placing of a few `defects' in an otherwise regular system, as e.g. the 12 pentagons of the otherwise graphene-like fullerene. Alexandrov's Theorem tells us that we can find a unique convex polyhedron in such a case. Several intending participants in this workshop have begun to investigate these polyhedra in smaller `toy' systems, but clearly need the help of experts on the mathematics and computer science. With this in mind, invitees will include members of the TU Berlin group.

We aim to construct Alexandrov polyhedra of fullerenes and correlate their metric properties with the physical properties of, e.g., fullerenes? This goal is more speculative than the others but with the right combination of researchers, rapid progress is a strong possibility.

6. The sixth and final objective is to determine how best to exploit the existing software for chemical structure generation and investigation. For the potential chemistry user there is a bewildering array of systems for graph generation, visualisation, calculation of invariants and conjecture making. These include:

fullgen, plantri, nauty, CaGe, fuigui, GrinVin, HouseOfGraphs, Vega, among others. The obvious practical need is for mechanisms for inter-package communication and interfacing, with at the least some common graph format(s)? It may be time to settle on a common platform; Sage is a strong candidate. Developers of many of the main codes and packages have indicated that they are keen to attend, as well as several Sage developers