Showing papers by "Andrey A. Dobrynin published in 2019"
TL;DR: A family of infinite families of transmission irregular trees of even order is constructed and it is shown that almost no graphs are transmission irregular.
23 citations
TL;DR: An infinite family of 3-connected cubic transmission irregular graphs is constructed and it is shown that almost no graphs are transmission irregular.
18 citations
TL;DR: An infinite family of 2-connected transmission irregular graphs is constructed and it is shown that almost no graphs are transmission irregular.
16 citations
07 Nov 2019
TL;DR: In this paper, the number of different vertex transmissions is defined as the Wiener complexity of a graph and the distance from a vertex to all the other vertices is the sum of distances from v to all vertices.
Abstract: Fullerenes are molecules that can be presented in the form of cage-like polyhedra, consisting only of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex v of a graph is a local graph invariant defined as the sum of distances from v to all the other vertices. The number of different vertex transmissions is called the Wiener complexity of a graph. Some calculation results on the Wiener complexity and the Wiener index of fullerene graphs of order n ≤ 232 and IPR fullerene graphs of order n ≤ 270 are presented. The structure of graphs with the maximal Wiener complexity or the maximal Wiener index is discussed, and formulas for the Wiener index of several families of graphs are obtained.
12 citations
TL;DR: The Wiener index as discussed by the authors is a topological index of a molecule defined as the sum of distances between all pairs of vertices in the chemical graph representing the non-hydrogen atoms in the molecule.
Abstract: The Wiener index is a topological index of a molecule, defined as the sum of distances between all pairs of vertices in the chemical graph representing the non-hydrogen atoms in the molecule. Hexagonal chains consist of hexagonal rings connected with each other by edges. This class of graphs contains molecular graphs of unbranched catacondensed benzenoid hydrocarbons. A segment of a chain is its maximal subchain with linear connected hexagons. Chains with segments of equal lengths can be coded by binary words. Formulas for the sums of Wiener indices of hexagonal chains of some families are derived and computational examples are presented.
10 citations
TL;DR: In this paper, the Wiener complexity of a graph is defined as the sum of distances from one vertex to all the other vertices in a graph, and the number of different vertex transmissions is calculated.
Abstract: Fullerenes are molecules in the form of cage-like polyhedra, consisting solely of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex $v$ of a graph is the sum of distances from $v$ to all the other vertices. The number of different vertex transmissions is called the Wiener complexity of a graph. Some calculation results on the Wiener complexity and the Wiener index of fullerene graphs of order $n \le 216$ are presented. Structure of graphs with the maximal Wiener complexity or the maximal Wiener index is discussed and formulas for the Wiener index of several families of graphs are obtained.
8 citations
01 Dec 2019
TL;DR: In this article, the authors determined how many annuli have certain properties for a fixed number of pentagons and showed that the number of annuli can be computed without overlapping on the plane.
Abstract: Pentahexagonal annuli are closed chains consisting of regular pentagons and hexagons. Such configurations can be easily recognized in various complex designs, in particular, in molecular carbon constructions. Results of computer enumeration of annuli without overlapping on the plane are presented for up to 18 pentagons and hexagons. We determine how many annuli have certain properties for a fixed number of pentagons. In particular, we consider symmetry, pentagon separation (the least ring-distance between pentagons), uniformity of pentagon distribution, and pentagonal thickness (the size of maximal connected part of pentagons) of annuli. Pictures of all annuli with the number of pentagons and hexagons up to 17 are presented (more than 1300 diagrams).
1 citations