The number of spanning trees in an (r,s)‐semiregular graph and its line graph

For a graph G, a “spanning tree” in G is a tree that has the same vertex set as G. The number of spanning trees in a graph (network) G, denoted by t(G), is an important invariant of the graph (network) with lots of decisive applications in many disciplines. In the article by Sato (Discrete Math. 2007, 307, 237), the number of spanning trees in an (r, s)‐semiregular graph and its line graph are obtained. In this article, we give short proofs for the formulas without using zeta functions. Furthermore, by applying the formula that enumerates the number of spanning trees in the line graph of an (r, s)‐semiregular graph, we give a new proof of Cayley's Theorem. © 2013 Wiley Periodicals, Inc.     

Microwave‐Induced Conversion of Acid Hydrazides to N,N′‐Diacylhydrazines in Solvent‐Free Conditions

A new and efficient method of converting acid hydrazides 1 to the corresponding N,N′‐diacylhydrazines 2 under microwave irradiation and in solvent‐free conditions has been described.     

Preparation and Deprotection of Aldehyde Dimethylhydrazones

Aldehydes were conveniently protected as dimethylhydrazones by stirring a mixture of the aldehyde, N,N‐dimethylhydrazine, anhydrous magnesium sulfate, and dichloromethane at room temperature. Azeotropic removal of water, formed during the course of the reaction, was not required because anhydrous magnesium sulfate functions as a water scavenger. Deprotection of aldehyde dimethylhydrazones was accomplished by stirring a mixture of the aldehyde dimethylhydrazone and aqueous glyoxylic acid at room temperature. The reaction time for the preparation and deprotection of aldehyde dimethylhydrazones varied with the structure of the aldehyde.     

One‐Pot,High‐Yielding Synthesis of Novel Dihydrothiazolo[3,2‐a]pyrimidinones

Reaction between 6‐(un)substituted‐2‐thiouracils and E‐ethyl 4‐bromocrotonate under basic conditions at room temperature is an easy, mild, high‐yielding, and regioselective method for the preparation of 7‐(un)substituted dihydrothiazolo[3,2‐a]pyrimidinone derivatives.     

Nickel Boride–Mediated Cleavage of 1,3-Dithiolanes: A Convenient Approach to Reductive Desulfurization

1,3-Dithiolanes are rapidly cleaved by nickel boride, generating corresponding hydrocarbons in excellent yields. The hydrogenolysis is rapid at room temperature and does not require protection from the atmosphere. Mild reaction conditions, simple workup, and good yields of pure products are some of the major advantages of the procedure.     

Oxidative Coupling Reaction of N,N‐Dialkylanilines with Cerium(IV) Ammonium Nitrate in the Solid State

Oxidative coupling reactions of N,N‐dialkylanilines with cerium(IV) ammonium nitrate can be achieved by grinding at room temperature in the absence of solvents.     

Stereocontrolled Transformations of trans α,β‐Aziridine Aldehydes toward Different Amino Hydroxylated Structures

The stereocontrolled condensation of methyl ketones to trans α,β‐aziridine aldehydes creates functionalized aldols, valuable precursors for different amino hydroxylated structures, such as amino alcohols, amino diols, α‐tetrahydrofuryl amides, and α‐furyl amides.     

α‐Cyanodithioic Acids and Their Corresponding Mono‐ and Dithiolate Salts as Building Blocks for the Synthesis of Novel Mercaptothiophenes

A novel and efficient method for the synthesis of mercaptothiophenes using cyanodithioic acids and their corresponding mono‐ and dithiolate salts as starting components is described.     

Efficient Synthesis of the Cyanofluoromethylated Ketones: Michael Reaction of the Cyanofluoromethyl Anion Species to Unsaturated Ketones

Abstract

Synthesis of cyanofluoromethylated ketones is discussed. The Michael addition of α‐cyano‐α‐fluoromethyl anion species to α,β‐unsaturated ketones is investigated.     

Synthesis of Some Novel α-Cyanoketene-N,S-acetals Derived from Secondary Aliphatic Amines and Their Use in Pyrazole Synthesis

New α-cyanoketene-N,S-acetals 2(a–g) and β-dialkylamine-α-cyanoacrylates 3(g–i) were synthesized in good to excellent yields by the reaction of ethyl 2-cyano-3,3-bis(methylthio)acrylate 1 with secondary aliphatic amines (i.e., N-methylalkyl- and N-ethylalkylamines), and pyrrolidine, in the presence of triethylamine, under reflux in ethanol, for 1–16 h, depending on the amine used. Five N-methylalkyl amines and pyrrolidine yielded exclusively mono-substituted N,S-acetals 2(a–f) in good yields. On the other hand, N-ethylbenzylamine gave a mixture of monosubstituted products including N,S-acetal 2g in 35% yield and the unexpected product ethyl 3-[benzyl(ethyl)amino]-2-cyanoacrylate 3g in 50% yield. N-Ethylcyclohexylamine and N-ethylbutylamine did not produce N,S-acetals and gave only the unexpected products ethyl 2-cyano-3-[cyclohexyl(ethyl)amino]acrylate 3h and ethyl 3-[butyl(ethyl)amino]-2-cyanoacrylate 3i in good yields. The α-cyanoketene-N,S-acetals 2(a–f), 2j, and 2k underwent cyclization with the binucleophile hydrazine in ethanol under reflux to afford substituted pyrazoles 4(a–f), 4j, and 4k in good yields.

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