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

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

Supplementary data and analysis for estimating walleye selectivity

This document has been issued as VIMS Data Report 61 and provides data tables and results of exploratory analyses conducted as part of the complete data analysis for Myers et al. 2014 published in the Transactions of the American Fisheries Society. Estimates of size- and sex-specific selectivity of fishing gear are important for making informed management decisions. We distinguish between capture selectivity – the relative catchability of the components of the population – and harvest selectivity, which is the combined effects of capture selectivity and the decision to retain or release a fish of a given population component. We used short-term recaptures from three extensive tagging programs in Minnesota and Wisconsin to estimate directly the size- and sex- specific selectivity of angling for captured and for harvested walleye Sander vitreus, and of spear fishing for harvested walleye. Estimates were obtained using generalized linear models with an information-theoretic approach to determining the significance of individual and interactive effects of length and sex on selectivity. The primary conclusions of this research are presented in Myers et al. 2014. Residual analyses for the models presented in the manuscript, results of unpublished exploratory analyses, and the complete data set used to conduct the analyses are presented in this supplementary document. Through this data report, interested readers can repeat the analyses conducted in Myers et al. 2014, as well as see the results of additional analyses not presented in the primary publication

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

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

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

In vitro degradation behavior and cytocompatibility of Mg–Zn–Zr alloys

Zinc and zirconium were selected as the alloying elements in biodegradable magnesium alloys, considering their strengthening effect and good biocompatibility. The degradation rate, hydrogen evolution, ion release, surface layer and in vitro cytotoxicity of two Mg–Zn–Zr alloys, i.e. ZK30 and ZK60, and a WE-type alloy (Mg–Y–RE–Zr) were investigated by means of long-term static immersion testing in Hank’s solution, non-static immersion testing in Hank’s solution and cell-material interaction analysis. It was found that, among these three magnesium alloys, ZK30 had the lowest degradation rate and the least hydrogen evolution. A magnesium calcium phosphate layer was formed on the surface of ZK30 sample during non-static immersion and its degradation caused minute changes in the ion concentrations and pH value of Hank’s solution. In addition, the ZK30 alloy showed insignificant cytotoxicity against bone marrow stromal cells as compared with biocompatible hydroxyapatite (HA) and the WE-type alloy. After prolonged incubation for 7 days, a stimulatory effect on cell proliferation was observed. The results of the present study suggested that ZK30 could be a promising material for biodegradable orthopedic implants and worth further investigation to evaluate its in vitro and in vivo degradation behavior