Some Reverse Topological Indices of Comet and Double Comet Graphs

A graph in which atoms are taken as vertices and bonds can be shown by edges is called a molecular graph. For such graphs we can find the indices showing their bioactivity as well as their physio-chemical properties. Industry is flourishing nowadays just because of the interdisciplinary study of different subjects. This gives us a way to understand the application of different subjects. Chemical graph theory is a mixture of chemistry and mathematics both play an important role in chemical graph theory. Chemistry provides a chemical compound and graph theory transform this chemical compound into a molecular graph which further studied by different aspects such as topological indices. In this article, we computed generalized Randić, general Zagreb, general sum-connectivity, ABC, GA, and indices of line graphs of the and nanotubes.”

Chemical graph theory has an important role in the development of chemical sciences. A graph is produced from certain molecular structure by means of applying several graphical operations. The local graph parameter is valency, which is defined for every vertex as the number associates with other vertices in a graph, for example an atom in a molecule. The demonstration of chemical networks and chemical compounds with the help of M-polynomials is a novel idea. The M-polynomial of different molecular structures help to compute several topological indices. A topological index is a numeric quantity that describes the whole structure of a molecular graph of the chemical compound and clarifies its physical features, chemical reactivates and boiling activities. In this paper we computed M-Polynomial and topological indices of TUHRC4 Graph, then we recovered numerous topological indices using the M-polynomials.

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.

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.

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.

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.

For a connected graph G and a, b is an element of R, the general degree-eccentricity index of G is defined as DEIa,b(G) = & sum;(v is an element of V(G) )d(G)(a)(v)ecc(G)(b)(v), where V(G) is the vertex set of G, d(G)(v) is the degree of a vertex v and ecc(G)(v) is the eccentricity of v in G, i.e. the maximum distance from v to another vertex of the graph. This index generalizes several well-known 'topological indices' of graphs such as the eccentric connectivity index. We characterize the unique unicyclic graphs with the maximum and the minimum general degree-eccentricity index among all n-vertex unicyclic graphs with fixed order, girth, and maximum degree for the cases a >= 1, b <= 0 and 0 <= a <= 1, b >= 0. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

A Topological index is a molecular structure descriptor of a molecular graph and it has many applications in real world issues. The Zagreb Indices based on degree of vertices of a graph are widely studied in chemical graph theory in over the past few years. Degree based Zagreb indices of undirected graphs {G_{m,n}}^M and G_n graphs are computed in this paper.

Topological index (TI) is a numerical invariant that helps to understand the natural relationship of the physicochemical properties of a compound in its primary structure. George Polya introduced the idea of counting polynomials in chemical graph theory and Winer made the use of TI in chemical compounds working on the paraffin's boiling point. The literature of the topological indices and counting polynomials of different graphs has grown extremely since that time. Metal-organic network (MON) is a group of different chemical compounds that consist of metal ions and organic ligands to represent unique morphology, excellent chemical stability, large pore volume, and very high surface area. Working on structures, characteristics, and synthesis of various MONs show the importance of these networks with useful applications, such as sensing of different gases, assessment of chemicals, environmental hazard, heterogeneous catalysis, gas and energy storage devices of excellent material, conducting solids, super-capacitors and catalysis for the purification, and separation of different gases. The above-mentioned properties and physical stability of these MONs become a most discussed topic nowadays. In this paper, we calculate the M-polynomials and various TIs based on these polynomials for two different MONs. A comparison among the aforesaid topological indices is also included to represent the better one.

Chemical graph theory is currently expanding the use of topological indices to numerically encode chemical structure. The prediction of the characteristics provided by the chemical structure of the molecule is a key feature of these topological indices. The concepts from graph theory are presented in a brief discussion of one of its many applications to chemistry, namely, the use of topological indices in quantitative structure-activity relationship (QSAR) studies and quantitative structure-property relationship (QSPR) studies. This study uses the M-polynomial approach, a newly discovered technique, to determine the topological indices of the medication fenofibrate. With the use of degree-based topological indices, we additionally construct a few novel degree based topological descriptors of fenofibrate structure using M-polynomial. When using M-polynomials in place of degree-based indices, the computation of the topological indices can be completed relatively quickly. The topological indices are also plotted. Using M-polynomial, we compute novel formulas for the modified first Zagreb index, modified second Zagreb index, first and second hyper Zagreb indices, SK index, S K 1 index, S K 2 index, modified Albertson index, redefined first Zagreb index, and degree-based topological indices.

In 1972, within a study of the structure-dependency of total π-electron energy (E), it was shown that E depends on the sum of squares of the vertex degrees of the molecular graph (later named first Zagreb index), and thus provides a measure of the branching of the carbon-atom skeleton. Topological indices are found to be very useful in chemistry, biochemistry and nanotechnology in isomer discrimination, structure–property relationship, structure-activity relationship and pharmaceutical drug design. In chemical graph theory, a topological index is a number related to a graph which is structurally invariant. One of the oldest most popular and extremely studied topological indices are well–known Zagreb indices. In a (molecular) graph G, the Zagreb topological index is equal to the sum of squares of the degrees of vertices of G and the Zagreb coindex is defined as the sum of a graph’s vertex degrees which is not adjacent. In this paper, we obtain the Zagreb coindex of four operations on graphs.

The combination of mathematical sciences, physical chemistry, and information sciences leads to a modern field known as cheminformatics. It shows a mathematical relationship between a property and structural attributes of different types of chemicals called quantitative-structures’ activity and qualitative-structures’ property relationships that are utilized to forecast the chemical sciences and biological properties, in the field of engineering and technology. Graph theory has originated a significant usage in the field of physical chemistry and mathematics that is famous as chemical graph theory. The computing of topological indices (TIs) is a new topic of chemical graphs that associates many physiochemical characteristics of the fundamental organic compounds. In this paper, we used the M-polynomial-based TIs such as 1st Zagreb, 2nd Zagreb, modified 2nd Zagreb, symmetric division deg, general Randi c ´ , inverse sum, harmonic, and augmented indices to study the chemical structures of pent-heptagonal nanosheets of V C 5 C 7 and H C 5 C 7 . An estimation among the computed TIs with the help of numerical results is also presented.

Graph theory plays substantial role in mathematics, chemistry, QSAR and physical sciences. The basic layout of the graph theoretic model is a molecular structure in which vertices of the graph correspond to atoms, and edges correspond to chemical bonds. The study of this graph model provides information about the chemical structure. A line graph has many useful applications in physical chemistry. M-polynomial is rich in producing closed forms of many degree-based topological indices which correlates chemical properties of the material under investigation. This polynomial is used in computing closed formulas of many degree-based topological invariants of the molecular structures. The molecular graph of carbon nanocones has a conical structure with a cycle of length k at its core and n layers of hexagons placed at the conical surface around its centre. In this study, we transformed the molecular structure of carbon nanocones into graph theoretic model and produced its line graph. Thereafter, we determined closed formulas for M-polynomials of line graphs of nanocones. We also recovered important topological degree-based indices of the line graph of nanocones. Moreover, we provide different graphs of topological indices and their relations with the parameters of the line graph of nanocones. These graphs depict the actual dependencies of the topological indices on the parameters of the carbon nanocones.

Expressions for the molecular topological features of silicon carbide compounds are essential for quantitative structure-property and structure-activity interactions. Chemical Graph Theory is a subfield of computational chemistry that investigates topological indices of molecular networks that correlate well with the chemical characteristics of chemical compounds. In the modern age, topological indices are extremely important in the study of graph theory. Topological indices are critical tools for understanding the core topology of chemical structures while examining chemical substances. In this article, compute the first and second k-Banhatti index, modified first and second k-Banhatti index, first and second k-hyper Banhatti index, first and second hyper Revan indices, first Revan vertex index, and third Revan index for Silicon Carbide SiC4-II [p, q] for all values of p and q

Chemical graph theory has become a significant part of the research. Topological indices are numerical descriptors used to determine physico-chemical properties of chemical compounds. In this work, the generalized redefined Zagreb index is defined. This index is applied on several topological indices viz., redefined Zagreb indices (ReZG1(G), ReZG2(G), ReZG3(G)), sum-connectivity Zagreb index (SCI(G)), hyper Zagreb index (HM(G)), harmonic index (H(G)), arithmetic-geometric index (AGI(G)), SK index (SK(G)) and SK1 index (SK1(G)). The above said indices are calculated for aromatic compounds such as boron triangular sheet, borophene chain and melem chain nanostructures. The obtained results are useful in understanding the behaviour of compounds under the study.