Graph Topological Indices Research Articles

Ensemble learning and graph topological indices for predicting physical properties of mental disorder drugs

In this paper, we use the ensemble machine learning technique to evaluate the strength of three supervised machine learning algorithms, namely, the random forest regression (RFR), support vector regression (SVR) and the gradient boosting regression (GBR) in the prediction of physical properties of mental disorder drugs with small dataset. The model was implemented on a dataset of neighborhood degree-based topological indices which served as predictor variables and physical properties of the drugs which served as target variables. To compute the neighborhood degree-based indices, we employed an algorithm that utilizes the canonical SmilES notations of the drugs. The ensemble method identifies the neighborhood third Zagreb index (NM3(G)) as an efficient predictor of boiling point, flash point and enthalpy of vaporization. The neighborhood Randic index (NR(G)) provides better prediction for molar refractivity, molar volume and polarizability. In the same vein, the neighborhood sum connectivity index (NSC(G)) is an efficient predictor of surface tension while the neighborhood reciprocal Randic index (NRR(G)) is most effective in the prediction of polar surface area. Furthermore, the comparison of the average performance between the ensemble method and the base models (RFR, SVR, GBR) over the neighborhood topological indices shows efficient performance of the individual models across multiple physical properties of mental disorder drugs, when using the neighborhood topological indices as the predictor or input feature. Overall, this research highlights the combination of three supervised machine learning models in an ensemble environment to mitigating the challenges associated with small datasets when applying machine learning models in QSPR analysis.

Study of Topological Behavior of Some Computer Related Graphs

Network theory is the study of graphs such as representing equilibrium relationships or unequal relationships between different objects. A network can be defined as a graph where nodes and / or margins have attributes (e.g. words). Topological index of a graph is a number that helps to understand its topology and a topological index is known as irregularity index if it is greater than zero and topological index of graph is equal to zero if and only if graph is regular. The irregularity indices are used for computational analysis of nonregular graph topological composition. In this paper, we aim to compute topological invariants of some computer related graph networks. We computed various irregularities indices for the graphs of OTIS swapped network OPa and Biswapped Networks Bsw(Pa).

Graph Energy Variants and Topological Indices in Platinum Anticancer Drug Design: Mathematical Insights and Computational Analysis with DFT and QTAIM

In this paper, we present a new approach that explores the application of graph energy variants in chemistry, specifically in the development of platinum anticancer drugs. While previous energy variants have been proposed without considering their direct relevance to chemistry, our study focuses on two key aspects. First, we investigate the correlation between seven degree-based and four distance-based topological indices and their corresponding energies in platinum anticancer drugs. Furthermore, we mathematically analyze the properties of these energies, establishing upper and lower bounds that can be generalized to other structures. Second, we examine the possibility of utilizing the energies of these topological indices as structural descriptors. Our research showcases promising results, suggesting potential improvements in the future manufacturing of anticancer drugs. In addition, we employ density functional theory D F T calculations to optimize the molecular structures of platinum anticancer drugs and identify local reactive sites using Fukui functions. Quantum theory of atoms in molecules Q T A I M was carried out at the bond critical point B C P , to reveal the nature of the intermolecular interactions in the investigated ten Pt anticancer drugs, especially, the nature of bonds between Pt atoms and their bond atoms. Overall, this study presents an innovative approach that bridges graph energies, topological indices, and DFT with the properties (physical and chemical) of platinum anticancer drugs, offering insights into their molecular properties and potential for enhanced drug design.

Relations Between the Energy and Topological Indices of a Graph

In this paper, we give various lower and upper bounds for the energy of graphs in terms of several topological indices of graphs: the first general multiplicative Zagreb index, the general Randić index, the general zeroth-order Randić index, the redefined Zagreb indices, and the atom-bond connectivity index. Moreover, we obtain new bounds for the energy in terms of certain graph invariants as diameter, girth, algebraic connectivity and radius.

Extremal quasi-unicyclic graphs with respect to vertex-degree function index

In this paper, the vertex-degree function index H f ( G ) is considered when function f ( x ) belongs to four classes of functions determined by the following properties: strictly convex versus strictly concave and strictly increasing versus strictly decreasing. Quasi-unicyclic graphs of given order (or of given order and fixed number of pendant vertices) extremal relatively to vertex-degree function index for these classes of functions are determined. These conditions are fulfilled by several topological indices of graphs.

Topological indices of intersection ideal graph of ℤn

Let n be a positive integer. The intersection graph of ring R=ℤn, is a graph whose vertices are non-empty and proper ideals of ℤn and two vertices are adjacent if and only if corresponding ideals have non-zero intersection and is denoted by G(ℤn). In this paper we compute some topological indices of this graph.

Topological Indices of Graphs from Vector Spaces

Topological indices are numbers that are applied to a graph and can be used to describe specific graph properties through algebraic structures. Algebraic graph theory is a helpful tool in a range of chemistry domains. Because it helps explain how the different symmetries of molecules and crystals affect their structure and dynamics, it is a powerful theoretical approach for forecasting both the common and uncommon characteristics of molecules. A topological index converts the chemical structure into a number and contributes a lot in chemical graph theory. In this article, we compute the Wiener index, Zagreb indexes, Wiener polynomial, Hyper-Wiener index, ABC index and eccentricity-based topological index of a nonzero component union graph from vector space.

On Independent Domination Topological Indices of Graphs

On Independent Domination Topological Indices of Graphs

Mathematical analysis of one-dimensional lead sulphide crystal structure using molecular graph theory

Nanoscience has progressed exponentially during the previous few decades, moving from bench science to complex applied technologies. Nanomaterials found their application in large number of industrial products which includes filters, paints, insulation, electronic components, lubricant additives, biomedical applications, etc. One such important nanomaterial is lead sulphide, an inorganic compound with the chemical formula PbS, and is one of the most common ore of Galena. The graphical depiction of the molecular structure determines how the atoms are organised in real three-dimensional space. Graphical representation encloses a detailed study of the physical and chemical properties of molecules for the calculation of degree and distance-based graph parameters. Such parameters are termed as topological indices of graph and are usually numerical parameters. The resolvability parameters of a graph are a highly innovative topic in which the entire structure is molded to obtain each atom's unique position. In this work, we consider some degree and distance-based graph parameters of an inorganic compound, namely, lead sulphide. In particular, we study its metric dimension and fault-tolerant metric dimension. We also obtain first, second, and third Zagreb index for PbS.

TOPOLOGICAL CHARACTERISTICS OF THE STRUCTURE OF COMPOSITE MATERIALS

The article discusses methods for modeling composite materials using graph theory. For this purpose, the method of structure-oriented and structure-invariant modeling of composite materials was analyzed. As a basis for such modeling, it is supposed to use structural descriptors ‒ quantities that describe the structure of the material at different scale levels, including the molecular one. Structure-oriented modeling of hierarchical systems, which, in particular, are composite materials, can be carried out on the basis of regression statistical modeling, which takes into account the possibility of implementing the previous structural level at the next one, and, in particular, the molecular level at the microscopic or mesoscopic level. A form of experimental-statistical models, which includes descriptors of several structural levels was proposed. A simplified approach, which takes into account the regularities of two levels: molecular and subsequent (micro- and mesoscopic) was considered. Examples and algorithms for constructing a representative graph for cross-linked and branched polymers, as well as silicate materials, were considered. It is shown that the representing graph of cross-linked polymers is infinite stochastic. An experimental procedure for constructing a discrete model based on microphotographs of a hardening binder was considered and implemented. For a quantitative description of this graph, an incremental scheme was used, as well as topological indices obtained as a result of the transformation of topological indices of graphs of low molecular weight compounds. For the purpose of transformation, there is a transition to probabilistic characteristics ‒ shares and average (normalized) values. The transformed topological indices are supposed to be applied in the statistical model of the composite material.

Brain Functional Topology Alteration in Right Lateral Occipital Cortex Is Associated With Upper Extremity Motor Recovery.

Stroke is a chief cause of sudden brain damage that severely disrupts the whole-brain network. However, the potential mechanisms of motor recovery after stroke are uncertain and the prognosis of poststroke upper extremity recovery is still a challenge. This study investigated the global and local topological properties of the brain functional connectome in patients with subacute ischemic stroke and their associations with the clinical measurements. A total of 57 patients, consisting of 29 left-sided and 28 right-sided stroke patients, and 32 age- and gender-matched healthy controls (HCs) were recruited to undergo a resting-state functional magnetic resonance imaging (rs-fMRI) study; patients were also clinically evaluated with the Upper Extremity Fugl-Meyer Assessment (FMA_UE). The assessment was repeated at 15 weeks to assess upper extremity functional recovery for the patient remaining in the study (12 left- 20 right-sided stroke patients). Global graph topological disruption indices of stroke patients were significantly decreased compared with HCs but these indices were not significantly associated with FMA_UE. In addition, local brain network structure of stroke patients was altered, and the altered regions were dependent on the stroke site. Significant associations between local degree and motor performance and its recovery were observed in the right lateral occipital cortex (R LOC) in the right-sided stroke patients. Our findings suggested that brain functional topologies alterations in R LOC are promising as prognostic biomarkers for right-sided subacute stroke. This cortical area might be a potential target to be further validated for non-invasive brain stimulation treatment to improve poststroke upper extremity recovery.

On the Uphill Zagreb Indices of Graphs

One of the tools, to research and investigation the structural dependence of various properties and some activities of chemical structures and networks is the topological indices of graphs. In this research work, we introduce novel indices of graphs which they based on the uphill degree of the vertices termed as uphill Zagreb topological indices. Exact formulae of these new indices for some important and famous families of graphs are established.

Computation of eccentric topological indices of zero-divisor graphs based on their edges

<abstract><p>The topological index of a graph gives its topological property that remains invariant up to graph automorphism. The topological indices which are based on the eccentricity of a chemical graph are molecular descriptors that remain constant in the whole molecular structure and therefore have a significant position in chemical graph theory. In recent years, various topological indices are intensively studied for a variety of graph structures. In this article, we will consider graph structures associated with zero-divisors of commutative rings, called zero-divisor graphs. We will compute the topological indices for a class of zero-divisor graphs of finite commutative rings that are based on their edge eccentricity. More precisely, we will compute the first and third index of Zagreb eccentricity, the eccentricity index of geometric arithmetic, the atomic bonding connectivity eccentricity index, and the eccentric harmonic index of the fourth type related to graphs constructed using zero-divisors of finite commutative rings $ \mathbb{Z}_{p^n}. $</p></abstract>

Investigation on Boron Alpha Nanotube by Studying Their M‐Polynomial and Topological Indices

Graph theory provides an effective tool such as graph polynomial and topological indices (TIs) to the chemist to analyze the different chemical structures. TIs are the numerical entity deducted from the molecular structure. TI helps to study the relationship between the physicochemical properties and structure of the chemical compound. In this article, we investigate the boron α‐nanotube by computing its M‐polynomial and then deducing its TI. Results are also shown by plotting the graphs.

Some New Upper Bounds for the Y‐Index of Graphs

In mathematical chemistry, the topological indices with highly correlation factor play a leading role specifically for developing crucial information in QSPR/QSAR analysis. Recently, there exists a new graph invariant, namely, Y‐index of graph proposed by Alameri as the sum of the fourth power of each and every vertex degree of that graph. The approximate range of the descriptors is determined by obtaining the bounds for the topological indices of graphs. In this paper, firstly, some upper bounds for the Y‐index on trees with several types of domination number are studied. Secondly, some new bounds are also presented for this index of graphs in terms of relevant parameters with other topological indices. Additionally, a new idea on bounds for the Y‐index by applying binary graph operations is computed.

Vertex-Based Topological Indices of Double and Strong Double Graph of Dutch Windmill Graph

A measurement of the molecular topology of graphs is known as a topological index, and several physical and chemical properties such as heat formation, boiling point, vaporization, enthalpy, and entropy are used to characterize them. Graph theory is useful in evaluating the relationship between various topological indices of some graphs derived by applying certain graph operations. Graph operations play an important role in many applications of graph theory because many big graphs can be obtained from small graphs. Here, we discuss two graph operations, i.e., double graph and strong double graph. In this article, we will compute the topological indices such as geometric arithmetic index GA , atom bond connectivity index ABC , forgotten index F , inverse sum indeg index ISI , general inverse sum indeg index ISI α , β , first multiplicative-Zagreb index PM 1 and second multiplicative-Zagreb index PM 2 , fifth geometric arithmetic index GA 5 , fourth atom bond connectivity index ABC 4 of double graph, and strong double graph of Dutch Windmill graph D 3 p .

On Topological Indices and QSPR Analysis of Drugs Used for the Treatment of Breast Cancer

A topological index of graph G is a numerical quantity which describes its topology. If it is applied to molecular structure of a chemical compounds then it reflects the theoretical properties of the chemical compounds. In this article, well-known degree based topological indices are applied on chemical structures of medicine used for treatment of breast cancer. Chemical structure is considered as graph, where elements are taken as vertices and bounds between them are taken as edges. Further, QSPR analysis of said topological indices are discussed and it is shown that these topological indices are highly correlated with the physical properties of chemical compounds used for the treatment of breast cancer.

Second Zagreb and Sigma Indices of Semi and Total Transformations of Graphs

The study of structure-property relations including the transformations of molecules is of utmost importance in correlations with corresponding physicochemical properties. The graph topological indices have been used effectively for such study and, in particular, bond-based indices play a vital role. The bond-additive topological indices of a molecular graph are defined as a sum of edge measures over all edges in which edge measures can be computed based on degrees, closeness, peripherality, and irregularity. In this study, we provide the mathematical characterization of the transformation of a structure that can be accomplished by the novel edge adjacency and incidence relations. We derive the exact expressions of bond type indices such as second Zagreb, sigma indices, and their coindices of total transformation and two types of semitransformations of the molecules which in turn can be used to characterize the topochemical and topostructural properties.

Topological indices of bipolar fuzzy incidence graph

Abstract The topological index of graph has a wide range of applications in theoretical chemistry, network design, data transmission, etc. In fuzzy graph settings, these topological indices have completely different definitions and connotations. In this work, we define new Wiener index and connectivity index for bipolar fuzzy incidence graphs, and obtain the characteristics of these indices by means of the definition of fuzzy membership functions. Furthermore, the interrelationship between Wiener index and connectivity index is considered.

General sum-connectivity index of unicyclic graphs with given diameter and girth

Topological indices of graphs have been studied due to their extensive applications in chemistry. We obtain lower bounds on the general sum-connectivity index [Formula: see text] for unicyclic graphs [Formula: see text] of given girth and diameter, and for unicyclic graphs of given diameter, where [Formula: see text]. We present the extremal graphs for all the bounds. Our results generalize previously known results on the harmonic index for unicyclic graphs of given diameter.