Total Domination in Graphs with Given Girth

A set S of vertices in a graph G without isolated vertices is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ _t (G) of G. In this paper, we establish an upper bound on the total domination number of a graph with minimum degree at least two in terms of its order and girth. We prove that if G is a graph of order n with minimum degree at least two and girth g, then γ _t (G) ≤ n/2 + n/g, and this bound is sharp. Our proof is an interplay between graph theory and transversals in hypergraphs. Michael A. Henning: Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

The Re(I) coordination chemistry of a series of pyrido[2,3-b]pyrazine-derived ligands: Syntheses,characterisation and crystal structures

Reaction of 2,3-diaminopyridine with one equivalent of a functionalised vicinal diketone, in ethanol, yields a series of ligands based upon the pyrido[2,3-b]pyrazine core. The ligands were characterised by ¹H, ¹³C–{¹H} NMR, MS and UV–Vis spectroscopy. Reaction of the ligands with one equivalent of {ReBr(CO)₅} gave a series of Re-Lⁿ complexes based upon the general formula fac-{ReBr(CO)₃(L)} (where L = pyrido[2,3-b]pyrazine-derived ligands, L¹–L⁶). Solution IR studies confirmed the retention of the facially capped, tri-carbonyl coordination geometry at rhenium, and ¹H NMR studies confirmed coordination of the ligand to Re(I). EI HR MS data were obtained for each complex confirming the proposed formulation and stoichiometry. Single crystal X-ray structures were obtained for three of the complexes (Re-L¹, Re-L², Re-L⁶), with each demonstrating that the ligands coordinate to Re(I) in a bidentate manner, via a four-membered chelate ring, which was unsymmetrical in the former two cases. The electronic absorption spectra of the complexes showed absorption into the visible region ca. 375–500 nm, (the complexes are orange-red in appearance). Following irradiation at 350–450 nm, the complexes display a solid-state broad emission peaking between 600–700 nm. The complexes were not sufficiently luminescent in solution to allow further investigation into the origin of this emission band, although with reference to related 1,8-naphthyridine complexes of Re(I) it is likely to incorporate significant ³MLCT character.     

A metacognitive-based instruction for Primary Four students to approach non-routine mathematical word problems

Many Primary Four students (fourth graders) in Singapore have difficulties initiating or persevering in the problem-solving process even though the curriculum has focused on problem solving since 1992. This study served to examine the role of metacognition in self-regulated problem solving. The study, a quasi-experimental pretest–posttest design involving a convenience sample of 63 students from two intact mixed-ability Primary Four classes, examined the impact of using a metacognitive scheme that focuses on the understanding and planning stages of Pòlya’s four-stage approach on students’ mathematical problem-solving behavior, performance and attitudes. The findings revealed that the metacognitive-based scheme had a positive impact on students’ understanding of the problem posed, solution planning, confidence in and personal control of problem-solving behavior and emotions. It had also helped them to initiate and persevere in the problem-solving process to achieve a higher level of problem-solving success. Limitations and instructional implications are discussed.     

Distinguishing-Transversal in Hypergraphs and Identifying Open Codes in Cubic Graphs

The open neighborhood N(v) of a vertex v in a graph G is the set of vertices adjacent to v in G. A graph is twin-free (or open identifiable) if every two distinct vertices have distinct open neighborhoods. A separating open code in G is a set C of vertices such that \({N(u) \cap C \neq N(v) \cap C}\) for all distinct vertices u and v in G. An open dominating set, or total dominating set, in G is a set C of vertices such that \({N(u) \cap C \ne N(v) \cap C}\) for all vertices v in G. An identifying open code of G is a set C that is both a separating open code and an open dominating set. A graph has an identifying open code if and only if it is twin-free. If G is twin-free, we denote by \({\gamma^{\rm IOC}(G)}\) the minimum cardinality of an identifying open code in G. A hypergraph H is identifiable if every two edges in H are distinct. A distinguishing-transversal T in an identifiable hypergraph H is a subset T of vertices in H that has a nonempty intersection with every edge of H (that is, T is a transversal in H) such that T distinguishes the edges, that is, \({e \cap T \neq f \cap T}\) for every two distinct edges e and f in H. The distinguishing-transversal number \({\tau_D(H)}\) of H is the minimum size of a distinguishing-transversal in H. We show that if H is a 3-uniform identifiable hypergraph of order n and size m with maximum degree at most 3, then \({20\tau_D(H) \leq 12n + 3m}\) . Using this result, we then show that if G is a twin-free cubic graph on n vertices, then \({\gamma^{\rm IOC}(G) \leq 3n/4}\) . This bound is achieved, for example, by the hypercube.     

Rapid detection of Mycobacterium tuberculosis cells by using microtip-based immunoassay

This paper describes a microtip-based approach of concentrating target analytes for a highly sensitive bioassay. As an example, rapid screening of bacterial whole cells is presented to detect Mycobacterium tuberculosis (MTB), a pathogenic bacterium for human tuberculosis (TB). The concentration and detection is performed with three sequential steps of (1) attracting bacterial whole cells in the vicinity of a microtip by alternating current electroosmotic flow; (2) capturing the cells onto the microtip by capillary action; (3) binding fluorophore-labeled polyclonal antibodies to the cells followed by fluorescence measurement (immunofluorescence). Through this mechanism, MTB cells have been detected to the concentration of 8,000 cells/mL within 10 min. This sensitivity is comparable to that of Ziehl–Neelsen smear microscopy, a common culture-free screening method for diagnosis of TB. For comparison, Escherichia coli O157:H7 cells have also been detected to the concentration of 30,000 cells/mL in the same way.     

Tight Lower Bounds on the Size of a Maximum Matching in a Regular Graph

In this paper we study tight lower bounds on the size of a maximum matching in a regular graph. For k ≥3, let G be a connected k-regular graph of order n and let α′(G) be the size of a maximum matching in G. We show that if k is even, then , while if k is odd, then . We show that both bounds are tight. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Effusing core at the center of A potential vortex

Experiments were carried out to measure the base pressure distribution of a flow field induced by a potential vortex with its axis normal to a stationary disk. The center base pressure coefficient of the vortex, C₀(0), was found to be proportional to Reynolds number from Re = 2.0 × 10³ to Re > 2.5 × 10⁴, where Re is based on the disk radius and azimuthal velocity at the disk edge. This behavior of C₀(0) is at variance with the experimental results of Phillips (Phys. Fluids, 27, 2215, 1984) and Khoo (M. Eng. Thesis, Natl. Univ. Singapore, 1984), which showed vastly different trends depending on Re. Plausible reasons are suggested for the apparent discrepancies observed. Finally, the extent of the effusing core at the center, r₁ (taken to be the radial position where departure from the outer potential flow took place), was found to be proportional to Re^−1/2 for all Re values considered.