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

Magneto-gyrotropic effects in semiconductor quantum wells (review)

Magneto-gyrotropic photogalvanic effects in quantum wells are reviewed. We discuss experimental data, results of phenomenological analysis and microscopic models of these effects. The current flow is driven by spin-dependent scattering in low-dimensional structures gyrotropic media resulted in asymmetry of photoexcitation and relaxation processes. Several applications of the effects are also considered.Comment: 28 pages, 13 figure

Relations for zeros of special polynomials associated to the Painleve equations

A method for finding relations for the roots of polynomials is presented. Our approach allows us to get a number of relations for the zeros of the classical polynomials and for the roots of special polynomials associated with rational solutions of the Painleve equations. We apply the method to obtain the relations for the zeros of several polynomials. They are: the Laguerre polynomials, the Yablonskii - Vorob'ev polynomials, the Umemura polynomials, the Ohyama polynomials, the generalized Okamoto polynomials, and the generalized Hermite polynomials. All the relations found can be considered as analogues of generalized Stieltjes relations.Comment: 17 pages, 5 figure

Cosmic microspheres in the Carboniferous deposits of the Usolka section (Urals foredeep)

© 2017Magnetite microspheres from the Carboniferous deposits of the Usolka reference section were studied by probe microanalysis, with comparison of the distributions of chemical elements and microspheres. The presence of microspheres in sedimentary strata is considered to be an additional factor for stratigraphic correlation between sedimentary sections. The microspheres are shown to be of cosmic nature. The Late Paleozoic paleoclimatic changes (extreme cooling) and biotic crises were caused by the periodical Solar System motion in the Galaxy, cosmic-dust fallout, and meteorite bombardments of the Earth