Numbers of cubic graphs

The numbers of unlabeled cubic graphs on p = 2n points have been found by two different counting methods, the best of which has given values for p ≦ 40.     

Distribution of subgraphs of random regular graphs

We obtain the asymptotic distribution of the number of copies of a fixed subgraph H in a random d‐regular graph, provided H is strictly balanced and d = d(n) is chosen so that the expected number of copies of H tends to infinity (but not too quickly), and the expected number of copies sharing edges with two other copies is bounded. The proof of asymptotic normality of the distribution uses a method of factorial moments for variables with unbounded means that was recently derived by the authors. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Eight stereoisomers of homonojirimycin from D-mannose

Although there are 32 6-azidoheptitols, there are only 16 homonojirimycin (HNJ) stereoisomers. Two epimeric azidoalditols derived from d-mannose allow the synthesis in water of eight stereoisomers of HNJ.     

Gradient-Constrained Minimum Networks. III. Fixed Topology

The gradient-constrained Steiner tree problem asks for a shortest total length network interconnecting a given set of points in 3-space, where the length of each edge of the network is determined by embedding it as a curve with absolute gradient no more than a given positive value m, and the network may contain additional nodes known as Steiner points. We study the problem for a fixed topology, and show that, apart from a few easily classified exceptions, if the positions of the Steiner points are such that the tree is not minimum for the given topology, then there exists a length reducing perturbation that moves exactly 1 or 2 Steiner points. In the conclusion, we discuss the application of this work to a heuristic algorithm for solving the global problem (across all topologies).     

The importance of including local correlation times in the calculation of inter-proton distances from NMR measurements: ignoring local correlation times leads to significant errors in the conformational analysis of the Glc alpha1-2Glc alpha linkage by NMR spectroscopy

The experimental determination of oligosaccharide conformations has traditionally used cross-linkage 1H-1H NOE/ROEs. As relatively few NOEs are observed, to provide sufficient conformational constraints this method relies on: accurate quantification of NOE intensities (positive constraints); analysis of absent NOEs (negative constraints); and hence calculation of inter-proton distances using the two-spin approximation. We have compared the results obtained by using 1H 2D NOESY, ROESY and T-ROESY experiments at 500 and 700 MHz to determine the conformation of the terminal Glc alpha1-2Glc alpha linkage in a dodecasaccharide and a related tetrasaccharide. For the tetrasaccharide, the NOESY and ROESY spectra produced the same qualitative pattern of linkage cross-peaks but the quantitative pattern, the relative peak intensities, was different. For the dodecasaccharide, the NOESY and ROESY spectra at 500 MHz produced a different qualitative pattern of linkage cross-peaks, with fewer peaks in the NOESY spectrum. At 700 MHz, the NOESY and ROESY spectra of the dodecasaccharide produced the same qualitative pattern of peaks, but again the relative peak intensities were different. These differences are due to very significant differences in the local correlation times for different proton pairs across this glycosidic linkage. The local correlation time for each proton pair was measured using the ratio of the NOESY and T-ROESY cross-relaxation rates, leaving the NOESY and ROESY as independent data sets for calculating the inter-proton distances. The inter-proton distances calculated including the effects of differences in local correlation times give much more consistent results.     

Hamiltonicity of random graphs produced by 2‐processes

Suppose that a random graph begins with n isolated vertices and evolves by edges being added at random, conditional upon all vertex degrees being at most 2. The final graph is usually 2‐regular, but is not uniformly distributed. Some properties of this final graph are already known, but the asymptotic probability of being a Hamilton cycle was not known. We answer this question along with some related questions about cycles arising in the process. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

3-star factors in random -regular graphs

The small subgraph conditioning method first appeared when Robinson and the second author showed the almost sure hamiltonicity of random d-regular graphs. Since then it has been used to study the almost sure existence of, and the asymptotic distribution of, regular spanning subgraphs of various types in random d-regular graphs and hypergraphs. In this paper, we use the method to prove the almost sure existence of 3-star factors in random d-regular graphs. This is essentially the first application of the method to non-regular subgraphs in such graphs.