Backbone Colouring of Graphs

Consider an undirected graph G and a subgraph of G, H. A q-backbone k-colouring of (G,H) is a mapping f: V(G) {1, 2, ..., k} such that G is properly coloured and for each edge of H, the colours of its endpoints differ by at least q. The minimum number k for which there is a backbone k-colouring of (G,H) is the backbone chromatic number, BBCq(G,H). It has been proved that backbone k-colouring of (G,T) is at most 4 if G is a connected C4-free planar graph or non-bipartite C5-free planar graph or Cj-free, j∈{6,7,8} planar graph without adjacent triangles. In this thesis we improve the results mentioned above and prove that 2-backbone k-colouring of any connected planar graphs without adjacent triangles is at most 4 by using a discharging method. In the second part of this thesis we further improve these results by proving that for any graph G with χ(G) ≥ 4, BBC(G,T) = χ(G). In fact, we prove the stronger result that a backbone tree T in G exists, such that ∀ uv ∈ T, |f(u)-f(v)|=2 or |f(u)-f(v)| ≥ k-2, k = χ(G). For the case that G is a planar graph, according to Four Colour Theorem, χ(G) = 4; so, BBC(G,T) = 4.

Histogram filtering as a tool in variational Monte Carlo optimization

Optimization of wave functions in quantum Monte Carlo is a difficult task because the statistical uncertainty inherent to the technique makes the absolute determination of the global minimum difficult. To optimize these wave functions we generate a large number of possible minima using many independently generated Monte Carlo ensembles and perform a conjugate gradient optimization. Then we construct histograms of the resulting nominally optimal parameter sets and "filter" them to identify which parameter sets "go together" to generate a local minimum. We follow with correlated-sampling verification runs to find the global minimum. We illustrate this technique for variance and variational energy optimization for a variety of wave functions for small systellls. For such optimized wave functions we calculate the variational energy and variance as well as various non-differential properties. The optimizations are either on par with or superior to determinations in the literature. Furthermore, we show that this technique is sufficiently robust that for molecules one may determine the optimal geometry at tIle same time as one optimizes the variational energy.

The Effects of Annealing URu2Si2 on the Resistivity and Meissner Effect

The enigmatic heavy fermion URu2Si2, which is the subject of this thesis, has attracted intensive theoretical and experimental research since 1984 when it was firstly reported by Schlabitz et al. at a conference [1]. The previous bulk property measurements clearly showed that one second order phase transition occurs at the Hidden Order temperature THO ≈ 17.5 K and another second order phase transition, the superconducting transition, occurs at Tc ≈ 1 K. Though twenty eight years have passed, the mechanisms behind these two phase transitions are still not clear to researchers. Perfect crystals do not exist. Different kinds of crystal defects can have considerable effects on the crystalline properties. Some of these defects can be eliminated, and hence the crystalline quality improved, by annealing. Previous publications showed that some bulk properties of URu2Si2 exhibited significant differences between as-grown samples and annealed samples. The present study shows that the annealing of URu2Si2 has some considerable effects on the resistivity and the DC magnetization. The effects of annealing on the resistivity are characterized by examining how the Residual Resistivity Ratio (RRR), the fitting parameters to an expression for the temperature dependence of the resistivity, the temperatures of the local maximum and local minimum of the resistivity at the Hidden Order phase transition and the Hidden Order Transition Width ∆THO change after annealing. The plots of one key fitting parameter, the onset temperature of the Hidden Order transition and ∆THO vs RRR are compared with those of Matsuda et al. [2]. Different media used to mount samples have some impact on how effectively the samples are cooled because the media have different thermal conductivity. The DC magnetization around the superconducting transition is presented for one unannealed sample under fields of 25 Oe and 50 Oe and one annealed sample under fields of 0 Oe and 25 Oe. The DC field dependent magnetization of the annealed Sample1-1 shows a typical field dependence of a Type-II superconductor. The lower critical field Hc1 is relatively high, which may be due to flux pinning by the crystal defects.