Spanning tree formulas and chebyshev polynomials

The Kirchhoff Matrix Tree Theorem provides an efficient algorithm for determiningt(G), the number of spanning trees of any graphG, in terms of a determinant. However for many special classes of graphs, one can avoid the evaluation of a determinant, as there are simple, explicit formulas that give the value oft(G). In this work we show that many of these formulas can be simply derived from known properties of Chebyshev polynomials. This is demonstrated for wheels, fans, ladders, Moebius ladders, and squares of cycles. The method is then used to derive a new spanning tree formula for the complete prismR _n (m) =K _m ×C _n . It is shown that $$2^{\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\left( {1 - \frac{1}{{r - 1}} + o\left( 1 \right)} \right)} $$ whereT _n (x) is then ^th order Chebyshev polynomial of the first kind.     

Molecular dynamics simulations of a nucleoside analogue of 1,4-dihydro-4-oxoquinoline-3-carboxylic acid synthesized as a potential antiviral agent: Conformational studies in vacuum and in water

Conformational analysis of nucleosides may have direct applications to the structure–activity relationship (SAR) studies and in the design of new drug candidates. Although conformational analysis may be accessed in many different ways, in this work it was performed using molecular dynamics (MD) simulation in order to study the dynamic behavior of a nucleoside derivative of 1,4-dihydro-4-oxoquinoline-3-carboxylic acid, synthesized by our group as a potential antiviral agent. The MD simulation was carried out during 10 ns in vacuum and in a box of water at two different temperatures (i.e., 300 and 600 K) using the AMBER force field. The in vacuum MD simulation results are in agreement with the crystallographic structure and with the DFT calculations of the nucleoside, revealing the anti conformer as the more stable one. The simulation in water, however, shows that both conformers may exist at 300 K, the temperature of the in vivo and in vitro assays, revealing that both the syn and anti conformers should be considered in a MD simulation study of the inhibitor–enzyme complex. Simulations are also in agreement with the NOE experiment, which shows that the anti conformer is the preferential one in DMSO-d6 solution at 298 K.     

Enhanced ethanol sensing properties of WO3 modified TiO2 nanorods

Pristine and WO₃ decorated TiO₂ nanorods (NRs) were synthesised to investigate n-n-type heterojunction gas sensing properties. TiO₂ NRs were fabricated via hydrothermal method on fluorine-doped tin oxide coated glass (FTO) substrates. Then, tungsten was sputtered on the TiO₂ NRs and thermally oxidised to obtain WO₃ nanoparticles. The heterostructure was characterised by X-ray diffraction (XRD), scanning electron microscopy (SEM), and energy-dispersive X-ray (EDX) spectroscopy. Fabricated sensor devices were exposed to VOCs such as toluene, xylene, acetone and ethanol, and humidity at different operation temperatures. Experimental results demonstrated that the heterostructure has better sensor response toward ethanol at 200 °C. Enhanced sensing properties are attributed to the heterojunction formation by decorating TiO₂ NRs with WO₃.     

Realizability of p-point graphs with prescribed minimum degree,maximum degree,and point-connectivity

In a recent paper, we gave a generalization of extremal problems involving certain graph-theoretic invariants. In that work, we defined a (p, Δ, δ, λ) graph as a graph having p points, maximum degree Δ, minimum degree δ, and line-connectivity λ. An arbitrary quadruple of integers (a, b, c, d) was called (p, Δ, δ, λ) realizable if there is a (p, Δ, δ, λ) graph with p = a, Δ = b, δ = c, and λ = d. In this work, we consider the more difficult case of (p, Δ, δ, κ) realizability, where κ is the point-connectivity. Necessary and sufficient conditions for a quadruple to be (p, Δ, δ, κ) realizable are derived.     

Quantitation of localized (31)P magnetic resonance spectra based on the reciprocity principle

There is a need for absolute quantitation methods in (31)P magnetic resonance spectroscopy, because none of the phosphorous-containing metabolites is necessarily constant in pathology. Here, a method for absolute quantitation of in vivo (31)P MR spectra that provides reproducible metabolite contents in institutional or standard units is described. It relies on the reciprocity principle, i.e., the proportionality between the B(1) field map and the map of reception strength for a coil with identical relative current distributions in receive and transmit mode. Cerebral tissue contents of (31)P metabolites were determined in a predominantly white matter-containing location in healthy subjects. The results are in good agreement with the literature and the interexamination coefficient of variance is better than that in most previous studies. A gender difference found for some of the (31)P metabolites may be explained by different voxel composition.     

Realizability of p-point graphs with prescribed minimum degree,maximum degree,and line connectivity

It is well known that certain graph-theoretic extremal questions play a central role in the study of communication network vulnerability. Herein we consider a generalization of some of the classical results in this area. We define a (p, Δ, δ, λ) graph as a graph having p points, maximum degree Δ, minimum degree Δ, and line connectivity λ. An arbitrary quadruple of integers (a, b, c, d) is called (p, Δ, δ, λ) realizable if there is a (p, Δ, δ, λ) graph with p = a, Δ = b, Δ = c, and λ = d. Necessary and sufficient conditions for a quadruple to be (p, Δ, δ, λ) realizable are derived.     

The strongest monotone degree condition for n-connectedness of a graph

It is shown that Bondy's degree condition for n-connectedness of a graph is the strongest monotone degree condition for forcibly n-connected graphic sequences.     

Phenolic compounds,essential oil composition,and antioxidant activity of Angelica purpurascens (Avé-Lall.) Gill

In this study, methanol extracts (MEs) and essential oil (EO) of Angelica purpurascens (Avé-Lall.) Gill obtained from different parts (root, stem, leaf, and seed) were evaluated in terms of antioxidant activity, total phenolics, compositions of phenolic compound, and essential oil with the methods of 2,2-azino-bis(3ethylbenzo-thiazoline-6-sulfonic acid (ABTS•⁺), 2,2-diphenyl-1-picrylhydrazil (DPPH•) radical scavenging activities, and ferric reducing/antioxidant power (FRAP), the Folin–Ciocalteu, liquid chromatography−tandem mass spectrometry (LC−MS/MS), and gas chromatography-mass spectrometry (GC−MS), respectively. The root extract of A. purpurascens exhibited the highest ABTS•⁺, DPPH•, and FRAP activities (IC₅₀: 0.05 ± 0.0001 mg/mL, IC₅₀: 0.06 ± 0.002 mg/mL, 821.04 ± 15.96 µM TEAC (Trolox equivalent antioxidant capacity), respectively). Moreover, EO of A. purpurascens root displayed DPPH• scavenging activity (IC₅₀: 2.95 ± 0.084 mg/mL). The root extract had the highest total phenolic content (438.75 ± 16.39 GAE (gallic acid equivalent), µg/mL)). Twenty compounds were identified by LC−MS/MS. The most abundant phenolics were ferulic acid (244.39 ± 15.64 μg/g extract), benzoic acid (138.18 ± 8.84 μg/g extract), oleuropein (78.04 ± 4.99 μg/g extract), and rutin (31.21 ± 2.00 μg/g extract) in seed, stem, root, and leaf extracts, respectively. According to the GC−MS analysis, the major components were determined as α-bisabolol (22.93%), cubebol (14.39%), α-pinene (11.63%), and α-limonene (9.41%) among 29 compounds. Consequently, the MEs and EO of A. purpurascens can be used as a natural antioxidant source.     

Effect of thermal history on conductivity and electrical relaxation in alkali silicate glasses

Electrical conductivity σ₀ and electric field relaxation measurements have been carried out as a function of thermal history for two alkali silicate glasses, Na₂O3SiO₂ and K₂O3SiO₂. Specimens of each glass with three different thermal histories, two of the anneal-and-quench type and one of the rate-cool type, were studied. The average structural or fictive temperature T_f of each of the specimens was characterized by measuring their indices of refraction. Effects of thermal history on σ₀ and its activation enthalpy $\text{H}{}_{\mathrm{\sigma }}{}^1$ were in accord with results of previous investigators. That is, for a given type of thermal history σ₀ was lower and $\text{H}{}_{\mathrm{\sigma }}{}^1$ higher the lower T_f. In addition it was found that for two specimens with the same T_f or index of refraction but different thermal histories the rate-cooled specimen exhibited a lower conductivity than the annealed-and-quenched specimen, in accord with the results of Ritland. The distribution of relaxation times τ_σ for decay of the electric field due to ionic migration was found to be due primarily to a distribution in the pre-exponential term ln $\text{τ}{}_{\mathrm{\sigma }}{}^1$ in the equation $\text{ln}\text{τ}{}_{\mathrm{\sigma }}\text{=}\text{ln}\text{τ}{}_{\mathrm{\sigma }}{}^1\text{+ H}{}^1\text{/RT}$; the distribution in $\text{H}{}^1$ was extremely narrow. Differences in thermal history caused small differences in the distribution of τ_σ, but no difference in the average activation enthalpy $\text{〈H}{}^1\text{〉}$ for τ_σ. From this result it appeared that the dependence of the conductivity activation enthalpy $\text{H}{}_{\mathrm{\sigma }}{}^1$ on thermal history was due to the effect of thermal history on the temperature dependence of the distribution in τ_σ.