Fair reception and Vizing's conjecture

In this paper we introduce the concept of fair reception of a graph which is related to its domination number. We prove that all graphs G with a fair reception of size γ(G) satisfy Vizing's conjecture on the domination number of Cartesian product graphs, by which we extend the well‐known result of Barcalkin and German concerning decomposable graphs. Combining our concept with a result of Aharoni, Berger and Ziv, we obtain an alternative proof of the theorem of Aharoni and Szabó that chordal graphs satisfy Vizing's conjecture. A new infinite family of graphs that satisfy Vizing's conjecture is also presented. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 45‐54, 2009     

Dominating direct products of graphs

An upper bound for the domination number of the direct product of graphs is proved. It in particular implies that for any graphs G and H, γ(G×H)?3γ(G)γ(H). Graphs with arbitrarily large domination numbers are constructed for which this bound is attained. Concerning the upper domination number we prove that Γ(G×H)?Γ(G)Γ(H), thus confirming a conjecture from [R. Nowakowski, D.F. Rall, Associative graph products and their independence, domination and coloring numbers, Discuss. Math. Graph Theory 16 (1996) 53-79]. Finally, for paired-domination of direct products we prove that γ_pr(G×H)?γ_pr(G)γ_pr(H) for arbitrary graphs G and H, and also present some infinite families of graphs that attain this bound.     

13C NMR of pterocarpans

¹³CNMR spectra of eight pterocarpans are reported Assignments are based on multiplicities and second-order features in off-resonance decoupled spectra and on chemical shifts in model compounds. Comparison of the degree of long-range (C, H) coupling in ¹H-coupled spectra is found useful     

Resolvent kernels of Green's function kernels and other finite-rank modifications of Fredholm and Volterra kernels

Many important Fredholm integral equations have separable kernels which are finite-rank modifications of Volterra kernels. This class includes Green's functions for Sturm-Liouville and other two-point boundary-value problems for linear ordinary differential operators. It is shown how to construct the Fredholm determinant, resolvent kernel, and eigenfunctions of kernels of this class by solving related Volterra integral equations and finite, linear algebraic systems. Applications to boundary-value problems are discussed, and explicit formulas are given for a simple example. Analytic and numerical approximation procedures for more general problems are indicated.This research was sponsored by the United States Army under Contract No. DAA29-75-C-0024.     

On Graphs Having Maximal Independent Sets of Exactly t Distinct Cardinalities

For a given positive integer t we consider graphs having maximal independent sets of precisely t distinct cardinalities and restrict our attention to those that have no vertices of degree one. In the situation when t is four or larger and the length of the shortest cycle is at least 6t ? 6, we completely characterize such graphs.     

MRI description of knee medial collateral ligament abnormalities in the absence of trauma: edema related to osteoarthritis and medial meniscal tears

A retrospective case series regarding the knees of 12 adult patients with MRI abnormalities of the medial collateral ligament (MCL), but without clinical history of trauma to the MCL, were collected and compared with six knee MR images from patients with clinical traumatic injuries to their MCLs. The MR images were studied for the extent of edema of the MCL, as well as other associated findings. Edema of the MCL on MRI could be found in three distinct categories of patients: (a) those with trauma to the MCL, which was an expected finding; (b) those without trauma but with medial compartment osteoarthritis; and (c) those without trauma but with degenerative medial meniscal tears. The clinical significance, if any, of the edema found in MCLs without trauma remains unclear. Atraumatic MCL edema may serve as a marker for medial knee compartment osteoarthritis or for a degenerative medial meniscal tear and should not be confused with traumatically induced MCL injuries, although this distinction can often be difficult to make based on MRI findings alone.     

Cell enumeration and visualisation by transmission electron microscopy of Lactobacillus rhamnosus treated with cinnamon (Cinnamomum zeylanicum B.) essential oil

The use of essential oils (EOs) in functional foods containing probiotic microorganisms must consider the antimicrobial activity of these oils against beneficial bacteria such as Lactobacillus rhamnosus. This study aimed to evaluate the sensitivity of L. rhamnosus cultures treated with cinnamon EO through viable cell counts and visualisation by transmission electron microscopy. Cinnamon EO at a concentration of 0.04% had a bacteriostatic activity after 2?h of incubation. Although slight alterations were detected in the cell structure, this concentration was considered to be bactericidal, since it led to a significant reduction in cell numbers after 24?h. On the other hand, cinnamon EO at a 1.00% concentration decreased cell counts by 3 log units after 2?h incubation and no viable cell count was detected after 24?h. Transmission electron microscopy indicated that cells treated with 1.00% cinnamon EO were severely damaged and presented cell membrane disruption and cytoplasmic leakage.     

Synthesis and Characterization of a Stable Nickelocenium Dication Salt

We report the synthesis and characterization of the nickelocenium cations [NiCp₂]⋅⁺ and [NiCp₂]²⁺ as their [F-{Al(OR^F)₃}₂]⁻ (Cp = C₅H₅; R^F=C(CF₃)₃) salts. Diamagnetic [NiCp₂]²⁺ represents the first example for the isolation of an unsubstituted parent metallocene dication. Both salts were generated by reacting neutral NiCp₂ with [NO]⁺[F-{Al(OR^F)₃}₂]⁻ in 1,2,3,4-tetrafluorobenzene (4FB). The salts were characterized by single crystal X-ray diffraction (XRD), indicating shorter metal-ligand bond lengths for the higher charged salt. Powder XRD shows the salts to be phase pure, cyclic voltammetry in 4FB gave quasi reversible redox waves at −0.44 (0→1) and +1.17 V (1→2) vs Fc/Fc⁺. The ¹H NMR of [NiCp₂]²⁺ is a singlet at 8.6 ppm, whereas paramagnetic [NiCp₂]⋅⁺ is significantly shifted upfield to −103.1 ppm.