Distance in Zigzag Polyhex Nanotubes

A topological index plays an important role in characterising various physical properties, biological activities, and chemical reactivities of a molecular graph. The Hosoya polynomial is used to evaluate the distance-based topological indices such as the Wiener index, hyper-Wiener index, Harary index, and Tratch – Stankevitch – Zefirov index. In the present study, we determine a closed form of the Hosoya polynomial for the third type of the chain hex-derived network of dimension n and derive the distance-based topological indices of the network with the help of their direct formulas and alternatively via using the obtained Hosoya polynomial. Finally, we graphically represent the computed distance-based topological indices and the Hosoya polynomial of the underlying network to comprehend their geometrical pattern. This study of the Hosoya polynomial and the corresponding indices can set the basis for more exploration into chain hex-derived networks and their properties.

Several bioactivities of chemical compounds in a molecular graph can be expected by using many topological descriptors. A topological descriptor is a numeric quantity which quantify the topology of a graph. By defining the metric on a graph related with a vector space, we consider this graph in the context of few topological descriptors, and quantify the Wiener index, hyper Wiener index, Reciprocal complimentary Wiener index, Schultz molecular topological index and Harary index. We also provide the graphical comparison of our results to describe the relationship and dependence of these descriptors on the involved parameters.

A topological index is a number generated from a molecular structure that signifies the molecule’s fundamental structural characteristics. This correlation between the index and various physical attributes is based on an algebraic quantity related to the chemical structure. Many topological indices, such as the Wiener index, first and second Zagreb indices, can be employed to determine various properties, including chemical activity, thermodynamic properties, physicochemical activity, and biological activity. Meanwhile, the p-subgroup graph of a group G is defined as a graph whose vertices represent the elements of the group, and two vertices are adjacent if and only if the order of the subgroup is a prime power. The main objective of this paper is to establish the general formula for certain topological indices, specifically the Wiener index, first Zagreb index, and second Zagreb index for the -subgroup graph associated with dihedral groups.

Topological indices are graph invariants computed usually by means of the distances or degrees of vertices of a graph. In chemical graph theory, a molecule can be modeled by a graph by replacing atoms by the vertices and bonds by the edges of this graph. Topological graph indices have been successfully used in determining the structural properties and in predicting certain physicochemical properties of chemical compounds. Wiener index is the oldest topological index which can be used for analyzing intrinsic properties of a molecular structure in chemistry. The Wiener index of a graph $G$ is equal to the sum of distances between all pairs of vertices of $G$. Recently, the entire versions of several indices have been introduced and studied due to their applications. Here we introduce the entire Wiener index of a graph. Exact values of this index for trees and some graph families are obtained, some properties and bounds for the entire Wiener index are established. Exact values of this new index for subdivision and $k$-subdivision graphs and some graph operations are obtained.

The advent of fuzzy sets, and consequently fuzzy graphs, has solved many problems in ambiguous and uncertain contexts. It is interesting and necessary to study the Wiener index in a cubic fuzzy graph that employs both fuzzy membership and interval-valued fuzzy membership at the same time. In this paper, the Wiener index in a cubic fuzzy graph is introduced as a cubic fuzzy number and some related results are described. The comparison between connectivity index and Wiener index, changes in Wiener index through deleting a node or an edge, and determining the Wiener index in some specific cubic fuzzy graphs have been the other topics studied in this research. In addition, the Wiener index is determined by mentioning concepts of the saturated cubic fuzzy cycle. In this review, the Wiener index is shown as a combination of classical and interval numbers. The results indicate that when some vertices are removed, the Wiener index may change. However, this change will not be exclusively related to both values. Finally, an application of the Wiener index is presented in the study of the properties of some monomer molecules.

Assume that is a finite group. The power graph of is a graph in which is its node set, where two different elements are connected by an edge whenever one of them is a power of the other. A topological index is a number generated from a molecular structure that indicates important structural properties of the proposed molecule. Indeed, it is a numerical quantity connected with the chemical composition that is used to correlate chemical structures with various physical characteristics, chemical reactivity, and biological activity. This information is important for identifying well-known chemical descriptors based on distance dependence. In this paper, we study Hosoya properties, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of power graphs of various finite cyclic and non-cyclic groups of order and , where and are prime numbers.

A fuzzy graph is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the fuzzy relation between any objects. The nature of fuzziness is favorable for any environment, which supports to predict the problem and solving it. Fuzzy graphs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. A topological index is a numerical quantity for the structural graph of the molecule and it can be represented through Graph theory. Moreover, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. A lot of topological indices are available in chemical-graph theory and H. Wiener proposed the first index to estimate the boiling point of alkanes called ‘Wiener index’. Many topological indices exist only in the crisp but it’s new to the fuzzy graph environment. The main aim of this paper is to define the topological indices in fuzzy graphs. Here, indices defined in fuzzy graphs are Modified Wiener index, Hyper Wiener index, Schultz index, Gutman index, Zagreb indices, Harmonic index, and Randić index with illustrations. Bounds for some of the indices are proved. The algorithms for distance matrix and MWI are shown. Finally, the application of these indices is discussed.

Introduction: Chemical Graph Theory or CGT for short, is a transdisciplinary field of Mathematics wherein graphs are used to represent chemical compounds. Under this representation, the atoms of a chemical compound are expressed as vertices, while the bonds connecting these atoms are expressed as edges. Once the graph representation of a chemical compound has been determined, graph-theoretic techniques can now be used to determine various topological indices associated with the chemical compound. Topological indices are invariants of a graph that have predictive power for the chemical properties of a chemical compound. In CGT, trees, being connected and acyclic, have been an essential class of graphs. This is because, most compounds have acyclic molecular structures. In the 2023 study by Gowtham and Husin, various reverse topological indices of a family of trees called bistar graphs have been determined. Objectives: Motivated by the work of Gowtham and Husin, we determine some reverse topological indices of another family of trees called comets and double comets. Double comets are natural extension of bistar graphs. We also give a certain computational application of our results on double comet graphs. Finally, we investigate the relationship between the reverse topological indices of bistar graphs and double comets. Methods: Several important graph-theoretic concepts were used to analyze some important properties of comets and double comets, leading to the computation of their reverse vertex degree based topological indices. Results: The following reverse vertex degree-based topological indices for comets and double comets were determined: Reverse sum-connectivity, First reverse Zagreb, Second reverse Zagreb, Reverse arithmetic-geometric, Reverse geometric-arithmetic, Reverse Sombor, and Reverse Nirmala. A computational application of the results to a certain chemical compound (2,2,4,4-Tetramethylpentane) or C9H20 were also demonstrated. Conclusions: The results of the paper provided an extension to an existing result on the reverse vertex degree-based topological indices of bistar graphs by considering double comet graphs. Several topological reverse vertex degree-based topological indices were also computed for comet graphs. The computed topological indices were also applied in determining some invariants of a certain chemical compound. For future studies, we recommend that reverse vertex degree based topological indices of other family of trees will be considered.

Chemical graph theory is a sub-branch of mathematical chemistry, assuming each atom of a molecule is a vertex and each bond between atoms as an edge. Owing to this theory, it is possible to avoid the difficulties of chemical analysis because many of the chemical properties of molecules can be determined and analyzed via topological indices. Due to these parameters, it is possible to determine the physicochemical properties, biological activities, environmental behaviours and spectral properties of molecules. Nowadays, studies on the zero divisor graph of Z_n via topological indices is a trending field in spectral graph theory. For a commutative ring R with identity, the prime ideal sum graph of R is a graph whose vertices are nonzero proper ideals of R and two distinctvertices I and J are adjacent if and only if I+J is a prime ideal of R. In this study the forgotten topological index and Wiener index of the prime ideal sum graph of Z_n are calculated for n=p^α,pq,p^2 q,p^2 q^2,pqr,p^3 q,p^2 qr,pqrs where p,q,r and s are distinct primes and a Sage math code is developed for designing graph and computing the indices. In the light of this study, it is possible to handle the other topological descriptors for computing and developing new algorithms for next studies and to study some spectrum and graph energies of certain finite rings with respect to PIS-graph easily.

The analysis of networks and graphs through topological descriptors carries out a useful role to derive their underlying topologies. This process has been widely used in biomedicine, cheminformatics, and bioinformatics, where assessments based on graph invariants have been made available for effectively communicating with the various challenging schemes. In the studies of quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs), graph invariants are used to approximate the biological activities and properties of chemical compounds. In this paper, we give the results related to the eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic index, eccentric atom-bond connectivity index, eccentric adjacency index, modified eccentric-connectivity index, eccentric distance sum, Wiener index, Harary index, hyper-Wiener index and degree distance index of a new graph operation named as “subdivision vertex-edge join” of three graphs.

V-Phenylenic nanotubes and nanotori are most comprehensively studied nanostructures due to widespread applications in the production of catalytic, gas-sensing and corrosion-resistant materials. Representing chemical compounds with M-polynomial is a recent idea and it produces nice formulas of degree-based topological indices which correlate chemical properties of the material under investigation. These indices are used in the development of quantitative structure-activity relationships (QSARs) in which the biological activity and other properties of molecules like boiling point, stability, strain energy etc. are correlated with their structures. In this paper, we determine general closed formulae for M-polynomials of V-Phylenic nanotubes and nanotori. We recover important topological degree-based indices. We also give different graphs of topological indices and their relations with the parameters of structures.

On topological indices for small RNA graphs

Aims: In this article, the drugs used in the treatment of Diabetic Nephropathy are analyzed using the topological indices, and a QSPR analysis of the chemical properties is done. Study Design: The degree based topological indices are used for this structural analysis of chemical compounds of these drugs. The graph structure of a chemical structure is derived from a chemical component by considering the atoms as vertices and the bonds connecting two atoms as edges, the edge-partitions are computed and used to obtain the topological indices. Results: The topological indices of these drugs are calculated and compared with the theoretical properties of the drugs. The diagrammatic visualization of the data is plotted in Box graph, Line graph and in scatter diagram. The theoretical values are compared with these topological indices and the regression equations are found. The relation between the properties of chemicals in terms of topological indices is given as regression equations. These equations contain vital information about the structural relationship and the numerical invariants and the properties in terms of numerical values and may be used for drug design. Conclusion: The computed topological indices are visualized in various aspects and the regression analysis among the chemical properties of the drugs and the topological indices is done. This degree-based QSPR analysis helps to design medicines for the diabetic nephropathy.

A topological index is a numeric quantity that is mathematically derived in a direct and unambiguous manner from the structural graph of a molecule. It has been found that many properties of a chemical compound are closely related to some topological indices of its molecular graph. Among topological indices, the Wiener index is probably the most important one. This index was introduced by the chemist H. Wiener (1947) about 60 years ago to demonstrate correlations between physicochemical properties of organic compounds and the topological structure of their molecular graphs. He defined his index as the sum of distances between any two carbon atoms in the molecules, in terms of carbon‐carbon bonds. The aim of this paper is to prepare some algorithms for computing distance matrix and Wiener index of zig‐zag and armchair polyhex nanotubes.

Network similarity measures have proven essential in the field of network analysis. Also, topological indices have been used to quantify the topology of networks and have been well studied. In this paper, we employ a new topological index which we call the Ediz eccentric connectivity index. We use this quantity to define network similarity measures as well. First, we determine the extremal value of the Ediz eccentric connectivity index on some network classes. Second, we compare the network similarity measure based on the Ediz eccentric connectivity index with other well-known topological indices such as Wiener index, graph energy, Randić index, the largest eigenvalue, the largest Laplacian eigenvalue, and connectivity eccentric index. Numerical results underpin the usefulness of the chosen measures. They show that our new measure outperforms all others, except the one based on Wiener index. This means that the measure based on Wiener index is still the best, but the new one has certain advantage to some extent.