The Generation of Fullerenes

We describe an efficient new algorithm for the generation of fullerenes. Our implementation of this algorithm is more than 3.5 times faster than the previously fastest generator for fullerenes -- fullgen -- and the first program since fullgen to be useful for more than 100 vertices. We also note a programming error in fullgen that caused problems for 136 or more vertices. We tabulate the numbers of fullerenes and IPR fullerenes up to 400 vertices. We also check up to 316 vertices a conjecture of Barnette that cubic planar graphs with maximum face size 6 are hamiltonian and verify that the smallest counterexample to the spiral conjecture has 380 vertices.Comment: 21 pages; added a not

Field theoretic approach to the counting problem of Hamiltonian cycles of graphs

A Hamiltonian cycle of a graph is a closed path that visits each site once and only once. I study a field theoretic representation for the number of Hamiltonian cycles for arbitrary graphs. By integrating out quadratic fluctuations around the saddle point, one obtains an estimate for the number which reflects characteristics of graphs well. The accuracy of the estimate is verified by applying it to 2d square lattices with various boundary conditions. This is the first example of extracting meaningful information from the quadratic approximation to the field theory representation.Comment: 5 pages, 3 figures, uses epsf.sty. Estimates for the site entropy and the gamma exponent indicated explicitl

Entropy of chains placed on the square lattice

We obtain the entropy of flexible linear chains composed of M monomers placed on the square lattice using a transfer matrix approach. An excluded volume interaction is included by considering the chains to be self-and mutually avoiding, and a fraction rho of the sites are occupied by monomers. We solve the problem exactly on stripes of increasing width m and then extrapolate our results to the two-dimensional limit to infinity using finite-size scaling. The extrapolated results for several finite values of M and in the polymer limit M to infinity for the cases where all lattice sites are occupied (rho=1) and for the partially filled case rho<1 are compared with earlier results. These results are exact for dimers (M=2) and full occupation (\rho=1) and derived from series expansions, mean-field like approximations, and transfer matrix calculations for some other cases. For small values of M, as well as for the polymer limit M to infinity, rather precise estimates of the entropy are obtained.Comment: 6 pages, 7 figure

Hamiltonian walks on Sierpinski and n-simplex fractals

We study Hamiltonian walks (HWs) on Sierpinski and $n$--simplex fractals. Via numerical analysis of exact recursion relations for the number of HWs we calculate the connectivity constant $\omega$ and find the asymptotic behaviour of the number of HWs. Depending on whether or not the polymer collapse transition is possible on a studied lattice, different scaling relations for the number of HWs are obtained. These relations are in general different from the well-known form characteristic of homogeneous lattices which has thus far been assumed to hold for fractal lattices too.Comment: 22 pages, 6 figures; final versio

Loop Model with Generalized Fugacity in Three Dimensions

A statistical model of loops on the three-dimensional lattice is proposed and is investigated. It is O(n)-type but has loop fugacity that depends on global three-dimensional shapes of loops in a particular fashion. It is shown that, despite this non-locality and the dimensionality, a layer-to-layer transfer matrix can be constructed as a product of local vertex weights for infinitely many points in the parameter space. Using this transfer matrix, the site entropy is estimated numerically in the fully packed limit.Comment: 16pages, 4 eps figures, (v2) typos and Table 3 corrected. Refs added, (v3) an error in an explanation of fig.2 corrected. Refs added. (v4) Changes in the presentatio

Exact Results for Hamiltonian Walks from the Solution of the Fully Packed Loop Model on the Honeycomb Lattice

We derive the nested Bethe Ansatz solution of the fully packed O($n$) loop model on the honeycomb lattice. From this solution we derive the bulk free energy per site along with the central charge and geometric scaling dimensions describing the critical behaviour. In the $n=0$ limit we obtain the exact compact exponents $\gamma=1$ and $\nu=1/2$ for Hamiltonian walks, along with the exact value $\kappa^2 = 3 \sqrt 3 /4$ for the connective constant (entropy). Although having sets of scaling dimensions in common, our results indicate that Hamiltonian walks on the honeycomb and Manhattan lattices lie in different universality classes.Comment: 12 pages, RevTeX, 3 figures supplied on request, ANU preprint MRR-050-9

Phenanthroindolizidine Alkaloids Isolated from <em>Tylophora ovata</em> as Potent Inhibitors of Inflammation, Spheroid Growth, and Invasion of Triple-Negative Breast Cancer

\ua9 2022 by the authors. Triple-negative breast cancer (TNBC), representing the most aggressive form of breast cancer with currently no targeted therapy available, is characterized by an inflammatory and hypoxic tumor microenvironment. To date, a broad spectrum of anti-tumor activities has been reported for phenanthroindolizidine alkaloids (PAs), however, their mode of action in TNBC remains elusive. Thus, we investigated six naturally occurring PAs extracted from the plant Tylophora ovata: O-methyltylophorinidine (1) and its five derivatives tylophorinidine (2), tylophoridicine E (3), 2-demethoxytylophorine (4), tylophoridicine D (5), and anhydrodehydrotylophorinidine (6). In comparison to natural (1) and for more-in depth studies, we also utilized a sample of synthetic O-methyltylophorinidine (1s). Our results indicate a remarkably effective blockade of nuclear factor kappa B (NFκB) within 2 h for compounds (1) and (1s) (IC50 = 17.1 \ub1 2.0 nM and 3.3 \ub1 0.2 nM) that is different from its effect on cell viability within 24 h (IC50 = 13.6 \ub1 0.4 nM and 4.2 \ub1 1 nM). Furthermore, NFκB inhibition data for the additional five analogues indicate a structure–activity relationship (SAR). Mechanistically, NFκB is significantly blocked through the stabilization of its inhibitor protein kappa B alpha (IκBα) under normoxic as well as hypoxic conditions. To better mimic the TNBC microenvironment in vitro, we established a 3D co-culture by combining the human TNBC cell line MDA-MB-231 with primary murine cancer-associated fibroblasts (CAF) and type I collagen. Compound (1) demonstrates superiority against the therapeutic gold standard paclitaxel by diminishing spheroid growth by 40% at 100 nM. The anti-proliferative effect of (1s) is distinct from paclitaxel in that it arrests the cell cycle at the G0/G1 state, thereby mediating a time-dependent delay in cell cycle progression. Furthermore, (1s) inhibited invasion of TNBC monoculture spheroids into a matrigel\uae-based environment at 10 nM. In conclusion, PAs serve as promising agents with presumably multiple target sites to combat inflammatory and hypoxia-driven cancer, such as TNBC, with a different mode of action than the currently applied chemotherapeutic drugs