Rainbow Connection Number of Graph Power and Graph Products

Rainbow connection number, rc(G), of a connected graph G is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same (note that the coloring need not be proper). In this paper we study the rainbow connection number with respect to three important graph product operations (namely the Cartesian product, the lexicographic product and the strong product) and the operation of taking the power of a graph. In this direction, we show that if G is a graph obtained by applying any of the operations mentioned above on non-trivial graphs, then rc(G) a parts per thousand currency sign 2r(G) + c, where r(G) denotes the radius of G and . In general the rainbow connection number of a bridgeless graph can be as high as the square of its radius 1]. This is an attempt to identify some graph classes which have rainbow connection number very close to the obvious lower bound of diameter (and thus the radius). The bounds reported are tight up to additive constants. The proofs are constructive and hence yield polynomial time -factor approximation algorithms.

Contributions to organic synthesis and reaction mechanisms

Development of new methods, leading to the first stereo-specific total synthesis of a steroid,viz equilenin, and of estrone and their derivatives and of several important synthones, useful for the preparation of physiologically active steroids, and the first conversion of an equilenane to estrane have been described. An account of the achievement of original syntheses of testosterone and its isomers and derivatives and degradation products, urinary steroids, terpenes and their important degradation products has been given. Mechanisms of Dieckmann cyclization, a novel dehydrogenation-addition reaction involving abietic acid and tetrachloro-o-benzoquinone, a rearrangement involving a substitution of cyclopentanone-2-carboxylic ester have been elucidated. An abnormaluv absorption exhibited by saturated 1,2-dicyano esters has been rationalized. Divergences in theord data of testosterone and 19-nortesto-sterone from their isomers have been explained by x-ray crystallographic studies of 8-isotestosterone, 8-iso-10-isotestosterone and 8-iso-10-iso-19-nortestosterone. A tentative explanation for the difference in their physiological activities has been suggested.

A novel amino acid racemization in vitamin b6-amino acid schiff base copper complexes: crystal structures of aquo (5'-phosphopyridoxylidene-dl-tyrosinato) copper(II) 3.5H2O and aquo (5'-phosphopyridoxylidene-dl-phenylalaninato) copper(II) 2.5H2O

A novel racemization observed in the Vitamin B6-amino acid Schiff base complexes, aquo (5'-phosphopyridoxylidene-l-tyrosinato) copper(II) and aquo (5'-phosphopyridoxylidene-l-phenylalaninato) copper(II) is described. The racemization taking place in solution under mild acidic conditions (pH 5-6) was confirmed by CD studies and the products were characterized by single crystal X-ray diffraction. The structures of both complexes show almost parallel orientation of the aromatic side chain and the pyridoxal II-system. The activation of the αCsingle bondH group due to the intermolecular II- interaction is probably the reason for the unusual racemization observed.

Characterization of Alkali Induced Formation of Lanthionine, Trisulfides, and Tetrasulfides from Peptide Disulfides using Negative Ion Mass Spectrometry

Peptide disulfides are unstable under alkaline conditions, resulting in the formation of products containing lanthionine and polysulfied linkages. Electrospray ionization mass spectrometry has been used to characterize major species obtained when cyclic and acyclic peptide disulfides are exposed to alkaline media. Studies on a model cyclic peptide disulfide (Boc - Cys - Pro - Leu - Cys - NHMe) and an acyclic peptide, oxidized glutathione, bis ((gamma)Glu Cys - Gly - COOH), are described. Disulfide cleavage reactions are initiated by the abstraction of (CH)-H-alpha or (CH)-H-beta protons of Cys residues, with Subsequent elimination of H2S or H2S2. The buildup of reactive thiol species which act on intermediates containing dehydroalanine residues, rationalizes the formation of lanthionine and polysulfide products. In the case of the cyclic peptide disulfide, the formation of cyclic products is facilitated by the intramolecular nature of the Michael addition reaction of thiols to the dehydroalanine residue. Mass spectral evidence for the intermediate species is presented by using alkylation of thiol groups as a trapping method. Mass spectral fragmentation in the negative ion mode of the peptides derived from trisulfides and tetrasulfides results in elimination of S-2. (J Am Soc Mass Spectrom 2009, 20, 783-791) (C) 2009 American Society for Mass Spectrometry.

Characterization of Alkali Induced Formation of Lanthionine, Trisulfides, and Tetrasulfides from Peptide Disulfides using Negative Ion Mass Spectrometry

Peptide disulfides are unstable under alkaline conditions, resulting in the formation of products containing lanthionine and polysulfied linkages. Electrospray ionization mass spectrometry has been used to characterize major species obtained when cyclic and acyclic peptide disulfides are exposed to alkaline media. Studies on a model cyclic peptide disulfide (Boc - Cys - Pro - Leu - Cys - NHMe) and an acyclic peptide, oxidized glutathione, bis ((gamma)Glu Cys - Gly - COOH), are described. Disulfide cleavage reactions are initiated by the abstraction of (CH)-H-alpha or (CH)-H-beta protons of Cys residues, with Subsequent elimination of H2S or H2S2. The buildup of reactive thiol species which act on intermediates containing dehydroalanine residues, rationalizes the formation of lanthionine and polysulfide products. In the case of the cyclic peptide disulfide, the formation of cyclic products is facilitated by the intramolecular nature of the Michael addition reaction of thiols to the dehydroalanine residue. Mass spectral evidence for the intermediate species is presented by using alkylation of thiol groups as a trapping method. Mass spectral fragmentation in the negative ion mode of the peptides derived from trisulfides and tetrasulfides results in elimination of S-2. (J Am Soc Mass Spectrom 2009, 20, 783-791) (C) 2009 American Society for Mass Spectrometry.

Studies on the Diels-Alder reaction of maleic anhydride and β-trans-ocimene

The structure of the by-product, obtained in the Diels-Alder condensation of maleic anhydride with β-trans-ocimene followed by distillation of the adduct formed, has been established as 2-isopropylidene-4-methyl-7-carboxy- ,3,3a,6,7,7a-hexahydroindanone (IVa) and the mechanism of its formation from the adduct (II) discussed. Some hitherto unreported reactions of the maleic anhydride adduct (II) and its derivatives are described. These throw light on the stereochemistry of the adduct and derived products.

A new chiral oxazoline derived from camphor and its conversion to (R)-2-methylalkanoic acids

1S,5R,7R)-(-)-10, 10-Dimethyl-3-ethyl-4-oxa--atricyclo[5.2.1.0(1,5)]dec-2-ene 2 was prepared in 95% yield from (1S)-1-amino-2-exo-hydroxyapocamphane 1. The chiral oxazoline could be alkylated (Lhttp://eprints.iisc.ernet.in/cgi/users/home?screen=EPrint::Edit&eprintid=31175&stage=core#tDA/THF/-78 degrees C/RX, RX = ethyl, n-propyl, n-butyl iodides or benzyl bromide) to 3 in 95% yield and > 95% diastereoselectivity, and the products hydrolysed to (R)-2-methylalkanoic acids 4 (43-47% yield, 93-98% e.e.). (C) 2000 Elsevier Science Ltd. All rights reserved.

Photochemical, photophysical, and theoretical studies on cyclobutanethiones..alpha.-Cleavage reactions

Photophysics and photochemistry of cyclobutanethiones 1-5 have been studied with the view to generalize the a-cleavage reactions of cyclobutanethiones. The above cyclobutanethiones possess a unit intersystem crossing efficiency from S1 to T1, a high self-quenching rate (-4 X lo9 M-' s-'), and a short triplet lifetime (<0.50 ws). Photolysis of 1-5 yields in benzene a product resulting from 1,3-transposition and in methanol two cyclic thioacetals.The origin of these products is traced to the triplet excited state. A mechanistic scheme involving a-cleavage as the primary photoprocess and diradicals and thiacarbenes as intermediates has been formulated to rationalize the formation of thioacetals and rearranged products. The proposed mechanistic scheme is supported by UHF MIND013 calculations performed on four model systems, cyclobutanethiones and 1,3-cyclobutanedithiones 18-21. These calculations indicate that formation of diradical is favored thermodynamically and kinetically for systems analogous to 19 and 21, while rearrangement to thiacarbene is likely only for those similar to 21.

Studies of phosphazenes. Part 33. Thermal rearrangement of alkoxy(p-methylphenoxy)cyclophosphazenes: a synthetic route to oxocyclophosphazanes, phosphazadienes and phosphaz-1-enes

The aryloxy(alkoxy)cyclotriphosphazenes N3P3(OR)6�m(OC6H4Me-p)n(R = Me, n= 1�3; R = Et or CH2Ph, n= 3) rearrange on heating to give trioxocyclotriphosphazanes; the di- and mono-methoxy derivatives, N3P3(OMe)6�n(OC6H4Me-p)n(n= 4 or 5), yield dioxophosphaz-1-enes and an oxophosphazadiene respectively. The 1H, 13C and 31P NMR data for the starting materials and the products are presented. No evidence has been found for partially rearranged products. The geometrical disposition of the aryloxy groups in the starting material is retained in the rearranged products. Some aspects of the mechanism of the thermal rearrangement are discussed.

Role of corundum-adapted strains of Bacillus polymyxa in the separation of hematite and alumina

Strains of Bacillus polymyxa, preadapted and grown in the presence of corundum, were found to be capable of the efficient separation of hematite from alumina. Results of rests peformed using binary hematite-corundum and ternary hematite-quartz-corundum mixtures in the presence of cells and metabolic products separated from the adapted bacterial culture indicated that more than 99% of the hematite could he efficiently separated through selective flocculation after desliming. It was found that alumina-specific bioproteins and other nonproteinaceous compounds were secreted by bacterial cells after adaptation to the mineral. The utility of this bioprocessing is demonstrated in the removal of iron from bauxite ores through selective flocculation in the presence of the adapted bacteria.

Selective biodissolution of calcium and iron from bauxite in the presence of Bacillus polymyxa

A bacterium Bacillus polymyxa was found to be capable of selective removal of calcium and iron from bauxite. The bioleached residue was found to be enriched in its alumina content with insignificant amounts of iron and calcium as impurities. The developed bio- process was found to be capable of producing a bauxite product which meets the specifica- tions as a raw material for the manufacture of alumina based ceramics and refractories. The role of bacterial cells and metabolic products in the selective dissolution of calcium (present as calcite) and iron (present as hematite and goethite) from bauxite was assessed and possi- ble mechanisms illustrated. The effect of different parameters such as sucrose concentra- tion, pH, pulp density and time on selective biodissolution was studied. It was observed that periodic decantation and replenishment of the leach medium was beneficial in improving the dissolution kinetics. Calcium removal involves chelation with bacterial exopolysaccha- tides and acidolysis by organic acid generation. Hematite could be solubilized through a reductive dissolution mechanism.

Algal biofuel production and mitigation potential in India

Energy and energy services are the backbone of growth and development in India and is increasingly dependent upon the use of fossil based fuels that lead to greenhouse gases (GHG) emissions and related concerns. Algal biofuels are being evolved as carbon (C)-neutral alternative biofuels. Algae are photosynthetic microorganisms that convert sunlight, water and carbon dioxide (CO2) to various sugars and lipids Tri-Acyl-Glycols (TAG) and show promise as an alternative, renewable and green fuel source for India. Compared to land based oilseed crops algae have potentially higher yields (5-12 g/m(2)/d) and can use locations and water resources not suited for agriculture. Within India, there is little additional land area for algal cultivation and therefore needs to be carried out in places that are already used for agriculture, e.g. flooded paddy lands (20 Mha) with village level technologies and on saline wastelands (3 Mha). Cultivating algae under such conditions requires novel multi-tier, multi-cyclic approaches of sharing land area without causing threats to food and water security as well as demand for additional fertilizer resources by adopting multi-tier cropping (algae-paddy) in decentralized open pond systems. A large part of the algal biofuel production is possible in flooded paddy crop land before the crop reaches dense canopies, in wastewaters (40 billion litres per day), in salt affected lands and in nutrient/diversity impoverished shallow coastline fishery. Mitigation will be achieved through avoidance of GHG, C-capture options and substitution of fossil fuels. Estimates made in this paper suggest that nearly half of the current transportation petro-fuels could be produced at such locations without disruption of food security, water security or overall sustainability. This shift can also provide significant mitigation avenues. The major adaptation needs are related to socio-technical acceptance for reuse of various wastelands, wastewaters and waste-derived energy and by-products through policy and attitude change efforts.

Metallacyclocumulenes: A Theoretical Perspective on the Structure, Bonding, and Reactivity

CONSPECTUS: Transition metals help to stabilize highly strained organic fragments. Metallacycles, especially unsaturated ones, provide much variety in this area. We had a sustained interest in understanding new C-C bond formation reactions affected by binuclear transition metal fragments Cp2M. One such study led to the exploration of the bimetallic C-C cleavage and coupled complexes, where the acetylide ligands bridge two metal atoms. The underlying M-C interaction in these complexes inspired the synthesis of a five-membered cyclocumulene complex, which opened a new phase in organometallic chemistry. The metallacyclocumulene produces a variety of C-C cleavage and coupled products including a radialene complex. Group 4 metallocenes have thus unlocked a fascinating chemistry by stabilizing strained unsaturated C4 organic fragments in the form of five-membered metallacyclocumulenes, metallacyclopentynes, and metallacycloallenes. Over the years, we have carried out a comprehensive theoretical study to understand the unusual stability and reactivity of these metallacycles. The unique (M-C-beta) interaction of the internal carbon atoms with the metal atom is the reason for unusual stability of the metallacycles. We have also shown that there is a definite dependence of the C-C coupling and cleavage reactions on the metal of metallacyclocumulenes. It demonstrates unexpected reaction pathways for these reactions. Based on this understanding, we have predicted and unraveled the stabilization factors of an unusual four-membered metallacycloallene complex. Indeed, our prediction about a four-membered heterometallacycle has led to an interesting bonding situation, which is experimentally realized. This type of M-C bonding is intriguing from a fundamental perspective and has great relevance in synthesizing unusual structures with interesting properties. In this Account, we first give a short prologue of what led to the present study and describe the salient features of the structure and bonding of the metallacyclocumulenes. The unusual reaction pathway of this metallacycle is explored next. Similar features of the metallacyclopentynes and metallacycloallenes are briefly mentioned. Then, we discuss the exploitation of the unique M-C bonding to design some exotic molecules such as a four-membered metallacycloallene complex. Our efforts to build a conceptual framework to understand these metallacycles and to exploit their chemistry continue.

Boxicity and cubicity of product graphs

The boxicity (cubicity) of a graph G is the minimum natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit cubes) in R-k. In this article, we give estimates on the boxicity and the cubicity of Cartesian, strong and direct products of graphs in terms of invariants of the component graphs. In particular, we study the growth, as a function of d, of the boxicity and the cubicity of the dth power of a graph with respect to the three products. Among others, we show a surprising result that the boxicity and the cubicity of the dth Cartesian power of any given finite graph is, respectively, in O(log d/ log log d) and circle dot(d/ log d). On the other hand, we show that there cannot exist any sublinear bound on the growth of the boxicity of powers of a general graph with respect to strong and direct products. (C) 2015 Elsevier Ltd. All rights reserved.