Doppelgangers and Lemke graphs

Let $G$ be a connected graph. A configuration of pebbles on $G$ is a function that assigns a nonnegative integer to each vertex. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. A configuration is solvable if after making pebbling moves any vertex can get at least one pebble. The pebbling number of $G$, denoted $\pi \left(G\right)$, is the smallest integer such that any configuration of $\pi \left(G\right)$ pebbles on $G$ is solvable. A graph has the two-pebbling property if after placing more than $2\pi \left(G\right)?q$ pebbles on $G$, where $q$ is the number of vertices with pebbles, there is a sequence of pebbling moves so that at least two pebbles can be placed on any vertex. A graph without the two-pebbling property is called a Lemke graph. Previously, an infinite family of Lemke graphs was shown to exist by subdividing edges of the original Lemke graph. In this paper, we introduce a new way to create infinite families of Lemke graphs based on adding vertices as well as subdividing edges. We also characterize the configurations that violate the two-pebbling property on these graphs and conjecture another infinite family of Lemke graphs that generalizes the original Lemke graph.     

Enantiopure vicinally trisubstituted all-cis-bis(hydroxymethyl)-cyclopentenols and their derivatives

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On the Laplace integral representation of multivariate Mittag‐Leffler functions in anomalous relaxation

In the given paper, a special method of representation of the Mittag‐Leffler functions and their multivariate generalizations in the form of the Laplace integrals is suggested. The method is based on the usage of the generalized multiplication Efros theorem. The possibilities of a new method are demonstrated on derivation of the integral representations for relaxation functions used in the anomalous dielectric relaxation in time domain. Copyright © 2016 John Wiley & Sons, Ltd.     

Synthesis of Novel Macrocyclic Ligands Containing Phosphoryl and Aminoacetal Fragments

Abstract

A method for the synthesis of novel phosphorylated aminoacetals was developed. The latter are involved in Mannich reaction as amine component with calix[4]resorcinol and formaldehyde to form tetrasubstituted macrocycles containing four acetal groups and four phosphonate (phosphine oxide) fragments on the “upper” rim of molecule with high yields.     

Synthesis of optically active spiro homo- and methanofullerenes

The selective synthesis of optically active spiro homofullerenes via reaction between [60]fullerene and cyclic diazoalkanes, generated in situ by oxidation of (−)-menthone hydrazone and d-(+)-camphor hydrazone in the presence of MnO₂, and the Pd(acac)₂-2PPh₃-4Et₃Al catalytic system was implemented for the first time. The optical rotations of the synthesized cycloadducts have been determined.