Some results on the Wiener index related to the \v{S}olt\'{e}s problem of graphs

The Wiener index, $W(G)$, of a connected graph $G$ is the sum of distances between its vertices. In 2021, Akhmejanova et al. posed the problem of finding graphs $G$ with large $R_m(G)= |\{v\in V(G)\,|\,W(G)-W(G-v)=m \in \mathbb{Z} \}|/ |V(G)|$. It is shown that there is a graph $G$ with $R_m(G) > 1/2$ for any integer $m \ge 0$. In particular, there is a regular graph of even degree with this property for any odd $m \ge 1$. The proposed approach allows to construct new families of graphs $G$ with $R_0(G) \rightarrow 1/2$ when the order of $G$ increases.Comment: 9 pages, 3 tables, 7 figure

Electrostatic Persistence Length of Semiflexible and Flexible Polyelectrolytes

ABSTRACT: I have calculated the dependence of the electrostatic persistence length on the Debye screening length for semiflexible, strongly and weakly charged flexible polyelectrolytes. For semiflexible and strongly charged flexible polyelectrolytes the electrostatic part of the chain persistence length is proportional to the Debye screening length. This result is obtained by evaluating the bending angle fluctuations and in the framework of the Gaussian variational principle. A polyelectrolyte chain with linear dependence of the electrostatic persistence length on the Debye screening length has a lower free energy than that of a chain with the Odijk-Skolnick-Fixman electrostatic persistence length. In the case of weakly charged chains the electrostatic persistence length has a sublinear dependence on the Debye screening length which is due to inverse logarithmic dependence of the linear charge density of the chain of electrostatic blobs on the Debye screening length. This result was derived by applying a coarse-graining procedure to an initially flexible polyelectrolyte chain by representing it as a chain of electrostatic blobs. The blob size and chain persistence length are then found self-consistently by minimizing the chain's variational free energy

Chains Are More Flexible Under Tension

The mechanical response of networks, gels, and brush layers is a manifestation of the elastic properties of the individual macromolecules. Furthermore, the elastic response of macromolecules to an applied force is the foundation of the single-molecule force spectroscopy techniques. The two main classes of models describing chain elasticity include the worm-like and freely-jointed chain models. The selection between these two classes of models is based on the assumptions about chain flexibility. In many experimental situations the choice is not clear and a model describing the crossover between these two limiting classes is therefore in high demand. We are proposing a unified chain deformation model which describes the force-deformation curve in terms of the chain bending constant K and bond length b. This model demonstrates that the worm-like and freely-jointed chain models correspond to two different regimes of polymer deformation and the crossover between these two regimes depends on the chain bending rigidity and the magnitude of the applied force. Polymer chains with bending constant K>1 behave as a worm-like chain under tension in the interval of the applied forces f ≤ KkBT/b and as a freely-jointed chain for f ≥ KkBT/b (kB is the Boltzmann constant and T is the absolute temperature). The proposed crossover expression for chain deformation is in excellent agreement with the results of the molecular dynamics simulations of chain deformation and single-molecule deformation experiments of biological and synthetic macromolecules

Programming temporal shapeshifting

Shapeshifting enables a wide range of engineering and biomedical applications, but until now transformations have required external triggers. This prerequisite limits viability in closed or inert systems and puts forward the challenge of developing materials with intrinsically encoded shape evolution. Herein we demonstrate programmable shape-memory materials that perform a sequence of encoded actuations under constant environment conditions without using an external trigger. We employ dual network hydrogels: in the first network, covalent crosslinks are introduced for elastic energy storage, and in the second one, temporary hydrogen-bonds regulate the energy release rate. Through strain-induced and time-dependent reorganization of the reversible hydrogen-bonds, this dual network allows for encoding both the rate and pathway of shape transformations on timescales from seconds to hours. This generic mechanism for programming trigger-free shapeshifting opens new ways to design autonomous actuators, drug-release systems and active implants

bis-Nitrile and bis-Dialkylcyanamide Platinum(II) Complexes as Efficient Catalysts for Hydrosilylation Cross-Linking of Siloxane Polymers

cis- and trans-Isomers of the platinum(II) nitrile complexes [PtCl2(NCR)2] (R = NMe2, N(C5H10), Ph, CH2Ph) were examined as catalysts for hydrosilylation cross-linking of vinyl-terminated polydimethylsiloxane and trimethylsilyl-terminated poly(dimethylsiloxane-co-ethylhydrosiloxane) producing high quality silicone rubbers. Among the tested platinum species the cis-complexes are much more active catalysts than their trans-congeners and for all studied platinum complexes cis-[PtCl2(NCCH2Ph)2] exhibits the best catalytic activity (room temperature, c = 1.0 10 4 mol/L, pot-life 60 min, curing 6 h). Although cis-[PtCl2(NCCH2Ph)2] is less active than the widely used Karstedt’s catalyst, its application for the cross-linking can be performed not only at room temperature (c = 1.0 10 4 mol/L), but also, more efficiently, at 80 C (c = 1.0 10 4–1.0 10 5 mol/L) and it prevents adherence of the formed silicone rubbers to equipment. The usage of the cis- and trans-[PtCl2(NCR)2] complexes as the hydrosilylation catalysts do not require any inhibitors and, moreover, the complexes and their mixtures with vinyl- and trimethylsilyl terminated polysiloxanes are shelf-stable in air. Tested catalysts do not form colloid platinum particles after the cross-linking.This project was supported by Federal Target Program (grant 14.576.21.0028). Andrey V. Vlasov and Vadim Yu. Kukushkin are much obliged to Saint Petersburg State University for a postdoctoral fellowship (12.50.1188.2014) and research grant (12.38.225.2014), correspondingly. The authors also express their gratitude to the Center of Thermal Analysis and Calorimetry (Saint Petersburg State University) for physicochemical measurements

Analytical “Bake-Off” of Whole Genome Sequencing Quality for the Genome Russia Project Using a Small Cohort for Autoimmune Hepatitis

A comparative analysis of whole genome sequencing (WGS) and genotype calling was initiated for ten human genome samples sequenced by St. Petersburg State University Peterhof Sequencing Center and by three commercial sequencing centers outside of Russia. The sequence quality, efficiency of DNA variant and genotype calling were compared with each other and with DNA microarrays for each of ten study subjects. We assessed calling of SNPs, indels, copy number variation, and the speed of WGS throughput promised. Twenty separate QC analyses showed high similarities among the sequence quality and called genotypes. The ten genomes tested by the centers included eight American patients afflicted with autoimmune hepatitis (AIH), plus one case’s unaffected parents, in a prelude to discovering genetic influences in this rare disease of unknown etiology. The detailed internal replication and parallel analyses allowed the observation of two of eight AIH cases carrying a rare allele genotype for a previously described AIH-associated gene (FTCD), plus multiple occurrences of known HLA-DRB1 alleles associated with AIH (HLA-DRB1-03:01:01, 13:01:01 and 7:01:01). We also list putative SNVs in other genes as suggestive in AIH influence

Super-soft and super-elastic dry gels

Molecular combs and bottlebrushes are a new class of polymer architecture allowing for anomalously low density of entanglements in polymer melts. The conformations and rheological properties of melts of these branched macromolecule composed of a flexible backbone and side chains densely tethered to it are investigated theoretically, experimentally and by computer simulations.1,2 We develop the rule for dialing in the desired value of the melt plateau modulus of these molecules as low as 1000 times below the conventional values for linear polymer melts and experimentally verify the validity of our theory. The theory also predicts that elastomers made from these melts should be super-elastic and reversibly stretch up to ten times more than elastomers made from linear polymers. Hybrid networks with both permanent and reversible bonds made with this novel architecture are predicted to be super-tough and self-healing. References W.F.M. Daniel, J. Burdynska, M. Vatankhah-Varnoosfaderani, K. Matyjaszewski, J. Paturej, M. Rubinstein, A.V. Dobrynin and S.S. Sheiko, Nature Materials, 2016, 15, 183-190. L.H Cai, T.E. Kodger, R.E. Guerra, A.F. Pegoraro, M. Rubinstein, and D.A. Weitz, Advanced Materials 2015, 27, 5132–5140

Bottlebrush Elastomers: A New Platform for Freestanding Electroactuation

Freestanding, single-component dielectric actuators are designed based on bottlebrush elastomers that enable giant reversible strokes at relatively low electric fields and altogether avoid preactuation mechanical manipulation. This materials design platform allows for independent tuning of actuator rigidity and elasticity over broad ranges without changing chemical composition, which opens new opportunities in soft-matter robotics

Genome-wide sequence analyses of ethnic populations across Russia

The Russian Federation is the largest and one of the most ethnically diverse countries in the world, however no centralized reference database of genetic variation exists to date. Such data are crucial for medical genetics and essential for studying population history. The Genome Russia Project aims at filling this gap by performing whole genome sequencing and analysis of peoples of the Russian Federation. Here we report the characterization of genome-wide variation of 264 healthy adults, including 60 newly sequenced samples. People of Russia carry known and novel genetic variants of adaptive, clinical and functional consequence that in many cases show allele frequency divergence from neighboring populations. Population genetics analyses revealed six phylogeographic partitions among indigenous ethnicities corresponding to their geographic locales. This study presents a characterization of population-specific genomic variation in Russia with results important for medical genetics and for understanding the dynamic population history of the world's largest country