The bandwidth problem for graphs and matrices—a survey

The bandwidth problem for a graph G is to label its n vertices v_i with distinct integers f(v_i) so that the quantity max{| f(v_i) ? f(v_i)| : (v_i v_j) ∈ E(G)} is minimized. The corresponding problem for a real symmetric matrix M is to find a symmetric permutation M' of M so that the quantity max{| i ? j| : m'_ij ≠ 0} is minimized. This survey describes all the results known to the authors as of approximately August 1981. These results include the effect on bandwidth of local operations such as refinement and contraction of graphs, bounds on bandwidth in terms of other graph invariants, the bandwidth of special classes of graphs, and approximate bandwidth algorithms for graphs and matrices. The survey concludes with a brief discussion of some problems related to bandwidth.     

Scandium(III) triflate as an efficient and reusable catalyst for synthesis of 1,5-benzodiazepine derivatives

2,3-Dihydro-1H-1,5-benzodiazepines have been synthesized in solvent-free conditions in excellent yield from o-phenylenediamines and ketones in the presence of a catalytic amount of Sc(OTf)₃. This method is a very easy, rapid, and high yielding reaction for the synthesis of 1,5-benzodiazepine derivatives.     

Comparison of AOAC 2005.06 LC official method with other methodologies for the quantitation of paralytic shellfish poisoning toxins in UK shellfish species

A refined version of the pre-column oxidation liquid chromatography with fluorescence detection (ox-LC-FLD) official method AOAC 2005.06 was developed in the UK and validated for the determination of paralytic shellfish poisoning toxins in UK shellfish. Analysis was undertaken here for the comparison of PSP toxicities determined using the LC method for a range of UK bivalve shellfish species against the official European reference method, the PSP mouse bioassay (MBA, AOAC 959.08). Comparative results indicated a good correlation in results for some species (mussels, cockles and clams) but a poor correlation for two species of oysters (Pacific oysters and native oysters), where the LC results in terms of total saxitoxin equivalents were found to be on average more than double the values determined by MBA. With the potential for either LC over-estimation or MBA under-estimation, additional oyster and mussel samples were analysed using MBA and ox-LC-FLD together with further analytical and functional methodologies: a post-column oxidation LC method (LC-ox-FLD), an electrophysiological assay and hydrophilic interaction liquid chromatography with tandem mass spectrometric detection. Results highlighted a good correlation among non-bioassay results, indicating a likely cause of difference was the under-estimation in the MBA, rather than an over-estimation in the LC results.     

The α-effect in cyclic secondary amines: new scaffolds for iminium ion accelerated transformations

Five-membered secondary amine heterocycles containing an α-heteroatom were prepared and shown to be ineffective as catalysts for the iminium ion catalysed Diels-Alder reaction between cinnamaldehyde and cyclopentadiene. Their six-membered counterparts proved to be highly active catalysts. In stark contrast, the catalytic activity observed when comparing the non α-heteroatom cyclic amines proline methyl ester and methyl pipecolinate showed the five-membered ring amine was significantly more active. Concurrent density functional theoretical calculations suggest a rationale for the observed trends in reactivity, highlighting that LUMO activation through an iminium ion intermediate plays a key role in catalytic activity.