Search Results (8)

The study of complex networks with topological indices has flourished in recent years. The aim of this paper is to study the limiting behaviors of Gutman and Schultz indices in random polygonal chains, whose graph-theoretic mathematical properties and their future applications have attracted the interest of scientists. By applying the concepts of symmetry and asymptotics as well as the knowledge of probability theory, we obtain explicit analytic expressions for the Gutman and Schultz indices of n random 2k-vertex chains and prove that they converge to a normal distribution, which contributes to a deeper understanding of the structural features of random polygonal chains and plays a crucial role in the study of the limiting behavior of topological indices and their applications. Full article

In this research paper, we investigate fundamental graph properties within the context of a simple connected graph denoted as G = (V, E). We introduce the concept of the rational Schultz index. In the context of this paper, our main objective is to calculate the rational Wiener index and rational Schultz index for a specific class of graphs. Our focus lies in the analysis and computation of these indices within this particular graph family. Full article

Two economic metrics demonstrate the actual, practical judgment of a country: on the one hand, the number of arriving tourists and overnight stays and their share within the GDP; on the other hand, the increase in FDI arriving in the country. The impact of macroenvironmental elements is becoming more and more intense in the life of each country. How a country deals with the issue of sustainability greatly influences the perception of a given destination among travelers. In other words, sustainability has become a serious element of a country’s image. Impressions of a country formulated by others build on the image the country authentically reflects about itself. This image, if successful, boosts countries’ competitiveness in all sectors. This study focuses on the liaison between tourism and a country’s brand. Researchers examined the body of networks and the interactions thereof. For this reason, the authors established a model to explain the relationship and coaction of a country’s brand and tourism. To build up research models, the authors used data from publicly available international public databases. The researchers detected areas where the improvement achieved increases the value of the country’s brand in addition to its competitiveness in tourism. Full article

In recent years, the research on complex networks has created a boom. The objective of the present paper is to study a random cyclooctatetraene chain whose graph-theoretic mathematical properties arose scientists’ interests. By applying the concept of symmetry and probability theory, we obtain the explicit analytical expressions for the variances of Schultz index, multiplicative degree-Kirchhoff index Gutman index, and additive degree-Kirchhoff index of a random cyclooctatetraene chain with n octagons, which plays a crucial role in the research and application of topological indices. Full article

Two famous topological indices, the Gutman index and Schultz index, are studied in this article. We mainly calculate the exact analytical formulae for the expected values of the Gutman index and Schultz index of a random regular polygonal chain with n regular polygons. Moreover, we determine the average values and the extremal values of the indices in regard to the set of all these regular polygonal chains. Full article

A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It is possible to determine several different properties, such as chemical activity, thermodynamic properties, physicochemical activity, and biological activity, using several topological indices, such as the geometric-arithmetic index, arithmetic-geometric index, Randić index, and the atom-bond connectivity indices. Consider $\mathcal{G}$ as a group and H as a non-empty subset of $\mathcal{G}$. The commuting graph $\mathcal{C}(\mathcal{G},H)$, has H as the vertex set, where $h_1,h_2\in H$ are edge connected whenever $h_1$ and $h_2$ commute in $\mathcal{G}$. This article examines the topological characteristics of commuting graphs having an algebraic structure by computing their atomic-bond connectivity index, the Wiener index and its reciprocal, the harmonic index, geometric-arithmetic index, Randić index, Harary index, and the Schultz molecular topological index. Moreover, we study the Hosoya properties, such as the Hosoya polynomial and the reciprocal statuses of the Hosoya polynomial of the commuting graphs of finite subgroups of $SL(2,\mathbb{C})$. Finally, we compute the Z-index of the commuting graphs of the binary dihedral groups. Full article

Introduction: People with depression often complain of dysfunction in cognitive processes, particularly attention. Pharmacotherapy is one of the most commonly used methods of treating depressive disorders and related attention difficulties. Patients also benefit from various forms of psychotherapy and frequently support themselves with alternative therapeutic methods. The purpose of this study was to examine whether a 15-min-long relaxation training session could improve the efficiency of attention and perceptiveness in individuals diagnosed with depressive disorders. Methods: Forty-two individuals participated in the study, including 20 individuals diagnosed with recurrent depressive disorder (rDD) and 22 healthy subjects (comparison group, CG). The so-called continuous performance test in the Polish version (Attention and Perceptiveness Test, APT) was applied in the study. In the first stage, the participants completed the 6/9 version of the APT test and then took part in a 15-min relaxation training session (autogenic training developed by the German psychiatrist Johannes Heinrich Schultz). The next step of the study was to perform APT again (parallel version—3/8). Results: The analyses showed statistically significant differences (p < 0.001) in the results obtained in the two versions of APT between the studied groups (rDD versus CG) in terms of the perceptual speed index. These differences were seen both before and after the introduction of the relaxation training. There was a statistically significant difference in the value of the perceptual speed index before and after the applied relaxation training for the subjects with depression (p = 0.004) and for the whole study group (p = 0.008). A significant correlation of illness symptom severity with decreased attentional efficiency was observed in the rDD group (perceptual speed index)—both before (r = −0.864; p < 0.001) and after the relaxation training (r = −0.785; p < 0.001). Conclusions: The continuous performance test (APT) is a reliable indicator of impaired attention efficiency among patients with depressive symptoms compared to healthy subjects. 15-min-long one-time relaxation exercise has a beneficial effect on attention efficiency measured by APT in people with depression. Full article

The study of topological indices – graph invariants that can be used for describing and predicting physicochemical or pharmacological properties of organic compounds – is currently one of the most active research fields in chemical graph theory. In this paper we study the Schultz index and find a relation with the Wiener index of the armchair polyhex nanotubes TUV C₆[2p; q]. An exact expression for Schultz index of this molecule is also found. Full article