Resistance distance local rules

In [D.J. Klein, Croat. Chem. Acta. 75(2), 633 (2002)] Klein established a number of sum rules to compute the resistance distance of an arbitrary graph, especially he gave a specific set of local sum rules that determined all resistance distances of a graph (saying the set of local sum rules is complete). Inspired by this result, we give another complete set of local rules, which is simple and also efficient, especially for distance-regular graphs. Finally some applications to chemical graphs (for example the Platonic solids as well as their vertex truncations, which include the graph of Buckminsterfullerene and the graph of boron nitride hetero-fullerenoid B ₁₂ N ₁₂) are made to illustrate our approach.     

On Unicycle Graphs with Maximum and Minimum Zeroth-order Genenal Randić Index

Let G be a graph and d _v denote the degree of the vertex v in G. The zeroth-order general Randić index of a graph is defined as R _α⁰(G) = ∑ _v∈V(G) d _v ^α where α is an arbitrary real number. In this paper, we obtained the lower and upper bounds for the zeroth-order general Randić index R _α⁰(G) among all unicycle graphs G of order n. We give a clear picture for R _α⁰(G) of unicycle graphs according to real number α in different intervals.     

Visual spectroscopy detection of triclosan

The azo coupling reaction with 2-aminonaphthalene-4,8-disulfonic acid (I) was used to develop a new cheap and rapid method of triclosan (II) determination in hygiene products. The calibration graph was linear in the range of 2.0−100 × 10⁻⁶ mol L⁻¹. The detection limit was 2.0 μmol L⁻¹.     

On the Randić Index

The Randić index of an organic molecule whose molecular graph G is defined as the sum of (d(u)d(v))^−1/2 over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G. In Delorme et al., Discrete Math. 257 (2002) 29, Delorme et al gave a best-possible lower bound on the Randić index of a triangle-free graph G with given minimum degree δ(G). In the paper, we first point out a mistake in the proof of their result (Theorem 2 of Delorme et al., Discrete Math. 257 (2002) 29), and then we will show that the result holds when δ(G)≥ 2.     

Outliers detection in the statistical accuracy test of a p<Emphasis Type="Italic">K</Emphasis><Subscript>a</Subscript> prediction

The regression diagnostics algorithm REGDIA in S-Plus is introduced to examine the accuracy of pK _a predicted with four programs: PALLAS, MARVIN, PERRIN and SYBYL. On basis of a statistical analysis of residuals, outlier diagnostics are proposed. Residual analysis of the ADSTAT program is based on examining goodness-of-fit via graphical diagnostics of 15 exploratory data analysis plots, such as bar plots, box-and-whisker plots, dot plots, midsum plots, symmetry plots, kurtosis plots, differential quantile plots, quantile-box plots, frequency polygons, histograms, quantile plots, quantile-quantile plots, rankit plots, scatter plots, and autocorrelation plots. Outliers in pK _a relate to molecules which are poorly characterized by the considered pK _a program. Of the seven most efficient diagnostic plots (the Williams graph, Graph of predicted residuals, Pregibon graph, Gray L–R graph, Index graph of Atkinson measure, Index graph of diagonal elements of the hat matrix and Rankit Q–Q graph of jackknife residuals) the Williams graph was selected to give the most reliable detection of outliers. The six statistical characteristics, F_exp,R²,R_P²,MEP,AIC{F_{\rm exp},R^{2},R_{\rm P}^{2},{\it MEP},{\it AIC}}, and s in pK _a units, successfully examine the specimen of 25 acids and bases of a Perrin’s data set classifying four pK _a prediction algorithms. The highest values F_exp,R²,R_P²{F_{\rm exp},R^{2},R_{\rm P}^{2}} and the lowest value of MEP and s and the most negative AIC have been found for PERRIN algorithm of pK _a prediction so this algorithm achieves the best predictive power and the most accurate results. The proposed accuracy test of the REGDIA program can also be extended to test other predicted values, as log P, log D, aqueous solubility or some physicochemical properties.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

On the Randić Index of Quasi-tree Graphs

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertex u and v in G. A graph G is called quasi-tree, if there exists such that G − u is a tree. In the paper, we give sharp lower and upper bounds on the Randić index of quasi-tree graphs. Mei Lu: Partially supported by NSFC (No. 10571105).     

On unicyclic conjugated molecules with minimal energies

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let U(k) be the set of all unicyclic graphs with a perfect matching. Let C _g(G) be the unique cycle of G with length g(G), and M(G) be a perfect matching of G. Let U ⁰(k) be the subset of U(k) such that g(G)≡ 0 (mod 4), there are just g/2 independence edges of M(G) in C _g(G) and there are some edges of E(G)\ M(G) in G\ C _g(G) for any G∈U ⁰(k). In this paper, we discuss the graphs with minimal and second minimal energies in U ^*(k) = U(k)\ U ⁰(k), the graph with minimal energy in U ⁰(k), and propose a conjecture on the graph with minimal energy in U(k).      

Some results on energy of unicyclic graphs with n vertices

A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Hou (J Math Chem 29:163–168, 2001) first considered the minimal energy for general unicyclic graphs. In this paper, we determine the unicyclic graphs with the minimal energy in U_n^l{\mathcal {U}_n^l} and the unicyclic graphs with the first forth smallest energy in U_n (n 3 13){\mathcal {U}_n\,(n\geq 13)} vertices.     

The Undecakisicosahedral Group and a 3-regular Carbon Network of Genus 26

Three projective special linear groups PSL(2,p), those with p = 5, 7 and 11, can be seen as p-multiples of tetrahedral, octahedral and icosahedral rotational point groups, respectively. The first two have already found applications in carbon chemistry and physics, as PSL(2,5) ≡ I is the rotation group of the fullerene C₆₀ and dodecahedrane C₂₀H₂₀, and PSL(2,7) is the rotation group of the 56-vertex all-heptagon Klein map, an idealisation of the hypothetical genus-3 “plumber’s nightmare” allotrope of carbon. Here, we present an analysis of PSL(2,11) as the rotation group of a 220-vertex, all 11-gon, 3-regular map, which provides the basis for a more exotic hypothetical sp ² framework of genus 26. The group structure and character table of PSL(2,11) are developed in chemical notation and a three dimensional (3D) geometrical realisation of the 220-vertex map is derived in terms of a punctured polyhedron model where each of 12 pentagons of the truncated icosahedron is connected by a tunnel to an interior void and the 20 hexagons are connected tetrahedrally in sets of 4.      

Flow-Injection Fluorimetric Determination of Thiamine in Pharmaceutical Preparations

 A new fluorimetric procedure for the determination of thiamine using flow injection analysis is proposed. The method is based on the derivatization reaction of the primary amine group with o-phthalaldehyde in the presence of 2-mercaptoethanol using fluorimetric detection. The calibration graph based on peak area was linear in the range 0.2–6 ng mL⁻¹. The detection limit was close to 0.1 ng mL⁻¹. The method was applied to the determination of the vitamin in commercial pharmaceutical preparations. Received March 31, 1999. Revision October 15, 1999.     

A note on the cyclical edge-connectivity of fullerene graphs

A fullerene graph is a 3-regular (cubic) and 3-connected spherical graph that has exactly 12 pentagonal faces and other hexagonal faces. The cyclical edge-connectivity of a graph G is the maximum integer k such that G cannot be separated into two components, each containing a cycle, by deletion of fewer than k edges. Došlić proved that the cyclical edge-connectivity of every fullerene graph is equal to 5. By using Euler’s formula, we give a simplified proof, mending a small oversight in Došlić’s proof. Further, it is proved that the cyclical connectivity of every fullerene graph is also equal to 5.     

Determination of norfloxacin in pharmaceuticals,human serum,and urine using a luminol—dissolved oxygen chemiluminescence system

A sensitive chemiluminescence method, based on the enhancive effect of norfloxacin on the reaction between luminol and dissolved oxygen in a flow injection system, was proposed for the determination of norfloxacin. The increment of the chemiluminiscence intensity was proportional to the concentration of norfloxacin, giving a calibration graph linear over the concentration from 0.4 ng mL⁻¹ to 400.0 ng mL⁻¹ (r ² = 0.9988) with the detection limit of 0.1 ng mL⁻¹ (3 × σ _noise). At the flow rate of 2.0 mL min⁻¹, a complete determination of norfloxacin, including sampling and washing, could be accomplished in 30 s with the relative standard deviation lower than 3.0 %. The proposed method was applied successfully to determine norfloxacin in pharmaceuticals, human urine, and serum. Possible mechanism of the reaction was also discussed.     

Simplified cloud point extraction for the preconcentration of ultra-trace amounts of gold prior to determination by electrothermal atomic absorption spectrometry

A simple and sensitive cloud point extraction method has been developed for the preconcentration of ultra-trace amounts of gold as a prior step to its determination by electrothermal atomic absorption spectrometry. It is based on the extraction of gold in hydrochloric acid medium using the non-ionic surfactant polyethyleneglycolmono-p-nonylphenylether (PONPE 7.5) without adding a chelating agent. The preconcentration of a 50 mL sample solution was thus enhanced by a factor of 200. The resulting calibration graph was linear in the range of 10–200 ng L⁻¹ with a correlation coefficient of 0.9993. The limit of detection (3s) obtained under optimal conditions was 2.0 ng L⁻¹. The relative standard deviation for 10 replicate determinations at a 100 ng L⁻¹ Au level was 3.6%. The method was applied to the ultra-trace determination of gold in water and copper samples.     

Maximum Randić index on Trees with <Emphasis Type="Italic">k</Emphasis>-pendant Vertices

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. Let T be a tree with n vertices and k pendant vertices. In this paper, we give a sharp upper bound on Randić index of T.     

Kinetic Determination of Iodide Based on its Effect on the Hydrogen Peroxide Oxidation of Triflupromazine

 A new selective, sensitive, and simple kinetic method is developed for the determination of trace amounts of iodide. The method is based on the catalytic effect of iodide on the reaction of triflupromazine (TFP) with H₂O₂. The reaction is followed spectrophotometrically by tracing the oxidation product at 498 nm within 1 min after addition of H₂O₂. The optimum reaction conditions are TFP (0.4 × 10⁻³ M), H₂SO₄ (1.0M), H₃PO₄ (2.0M), and H₂O₂ (1.6M) at 30°C. Following this procedure, iodide can be determined with a linear calibration graph up to 4.5 ng ċ cm⁻³ and a detection limit of 0.04 ng ċ cm⁻³, based on the 3 S_b criterion. The method can also be applied to the determination of iodate and periodate ions. Determination of as little as 0.2, 1.0, 2.0, and 4.0 ng ċ cm⁻³ of I⁻, IO₃ ^-, or IO₄ ^- in aqueous solutions gave an average recovery of 98% with relative standard deviations below 1.6% (n = 5). The method was applied to the determination of iodide in Nile river water and ground waters as well as in various food samples after alkaline ashing treatment. The method is compared with other catalytic spectrophotometric procedures for iodide determination.     

Minimum general sum-connectivity index of unicyclic graphs

The general sum-connectivity index of a graph G is defined as χ _α (G) = ∑_edges (d _u + d _v ) ^α , where d _u denotes the degree of vertex u in G and α is a real number. In this report, we determine the minimum and the second minimum values of the general sum-connectivity indices of n-vertex unicyclic graphs for non-zero α ≥ −1, and characterize the corresponding extremal graphs.     

Kinetic Determination of Iodide Based on its Effect on the Hydrogen Peroxide Oxidation of Triflupromazine

Summary.  A new selective, sensitive, and simple kinetic method is developed for the determination of trace amounts of iodide. The method is based on the catalytic effect of iodide on the reaction of triflupromazine (TFP) with H₂O₂. The reaction is followed spectrophotometrically by tracing the oxidation product at 498 nm within 1 min after addition of H₂O₂. The optimum reaction conditions are TFP (0.4 × 10⁻³ M), H₂SO₄ (1.0M), H₃PO₄ (2.0M), and H₂O₂ (1.6M) at 30°C. Following this procedure, iodide can be determined with a linear calibration graph up to 4.5 ng ċ cm⁻³ and a detection limit of 0.04 ng ċ cm⁻³, based on the 3 S_b criterion. The method can also be applied to the determination of iodate and periodate ions. Determination of as little as 0.2, 1.0, 2.0, and 4.0 ng ċ cm⁻³ of I⁻, IO₃ ^-, or IO₄ ^- in aqueous solutions gave an average recovery of 98% with relative standard deviations below 1.6% (n = 5). The method was applied to the determination of iodide in Nile river water and ground waters as well as in various food samples after alkaline ashing treatment. The method is compared with other catalytic spectrophotometric procedures for iodide determination. Received January 19, 2001. Accepted (revised) March 12, 2001