Molecular Graph Indexes for Assessing Heterogeneity of Chemical Compounds

The quantitative structures activity relationships (QSAR) and quantitative structures property relationships (QSPR) between the chemical compounds are studied with the help of topological indices (TI’s) which are the fixed real numbers directly linked with the molecular graphs. Gutman and Trinajstic (1972) defined the first degree based TI to measure the total π-electrone energy of a molecular graph. Recently, Ali and Trinajstic (2018) restudied the connection based TI’s such as first Zagreb connection index, second Zagreb connection index and modified first Zagreb connection index to find entropy and accentric factor of the octane isomers. In this paper, we study the modified second Zagreb connection index and modified third Zagreb connection index on the T-sum (molecular) graphs obtained by the operations of subdivision and product on two graphs. At the end, as the applications of the obtained results for the modified Zagreb connection indices of the T-sum graphs of the particular classes of alkanes are also included. Mainly, a comparision among the Zagreb indices, Zagreb connection indices and modified Zagreb connection indices of the T-sum graphs of the particular classes of alkanes is performed with the help of numerical tables, 3D plots and line graphs using the statistical tools.

The Padmakar-Ivan (PI) index of a graph G is defined as PI(G) = ∑(n eu (e|G)+ n ev (e|G)), where n eu (e|G) is the number of edges of G lying closer to u than to v, n ev (e|G) is the number of edges of G lying closer to v than to u and summation goes over all edges of G. In this paper, we first compute the PI index of a class of pericondensed benzenoid graphs consisting of n rows, n ≤ 3, of hexagons of various lengths. Finally, we prove that for any connected graph G with exactly m edges, PI(G) ≤ m(m-1) with equality if and only if G is an acyclic graph or a cycle of odd length. 1 . Here, we consider a new topological index, named the Padmakar-Ivan index, which is abbreviated as the PI index 2-17 . This newly proposed topological index, differ from the Wiener index 18 , the oldest topological index for acyclic (tree) molecules. We now describe some notations which will be adhered to throughout. Benzenoid systems (graph representations of benzenoid hydrocarbons) are defined as finite connected plane graphs with no cut-vertices, in which all interior regions are mutually congruent regular hexagons. More details on this important class of molecular graphs can be found in the book of Gutman and Cyvin 19 and in the references cited therein. Let G be a simple molecular graph without directed or multiple edges and without loops, the vertex and edge-shapes of which are represented by V(G) and E(G), respectively. The graph G is said to be connected if for every pair of vertices x and y in V(G) there exists a path between x and y. In this paper we only consider connected graphs. If e is an edge of G, connecting the vertices u and v then we write e=uv. The number of vertices of G is denoted by n. The distance between a pair of vertices u and w of G is denoted by d(u,w). We now define the PI index of a graph G. To do this, suppose that e = uv and introduce the quantities neu(e|G) and nev(e|G). neu(e|G) is the number of edges lying closer to vertex u than to vertex v, and nev(e|G) is the number of edges lying closer to vertex v than to vertex u. Then PI(G) = ∑(neu(e|G) + nev(e|G)), where the summation goes over all edges of G. Edges equidistant from both ends of the edge e = uv are not counted and the number of such edges is denoted by N(e). To clarify this, for every vertex u and any edge f = zw of graph G, we define d(f,u) = Min{d(u,w),d(u,z)}. Then f is equidistant from both ends of the edge e = uv if d(f,u) = d(f,v). In a series of papers, Khadikar and coauthors 2-17 defined and then computed the PI index of some chemical graphs. The present author 20 computed the PI index of a zig-zag polyhex nanotube. In this paper we continue this study to prove an important result concerning the PI index and find an exact expression for the PI index of some other chemical graphs. Our notation is standard and mainly taken from the literature. 21,22

A chemical compound can be represented as a chemical graph. A topological index of a (chemical) graph is a numeric value of a graph which characterize its topology and is usually graph invariant. The Zagreb indices, Randić index and sum-connectivity indices are useful in the study of anti-inflammatory activities, boiling point, molecular complexity heterosystems of certain chemical instances, and in elsewhere. In this paper, we calculate the mentioned topological indices of some infinite classes of prophyrin dendrimers.

The Euler-Sombor index is a vertex-degree-based topological index defined for a graph G = (V,E) as where du and dv denote the degrees of adjacent vertices u and v. In prior work [6], the chemical trees and unicyclic graphs with the minimum Euler-Sombor indices were characterized, prompting the search for extremal values over all chemical graphs. This paper extends that work by determining the first three minimum Euler-Sombor indices among chemical bicyclic graphs and the first seven among chemical tricyclic graphs. Furthermore, we prove that the minimum Euler-Sombor index for both unicyclic and bicyclic chemical graphs is attained when the maximum degree is at most 4, thereby conclusively establishing the minimum values for these graph classes.

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.

The sum‐connectivity (SC) index of a graph is defined as , where Θ μ denotes the vertex degree of μ in . In this paper, the fourth largest value of SC index for the chemical unicyclic graphs of order n ≥ 7 is determined.

Let G be a molecular graph. The distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. In this paper, we consider some infinite families of 3-fence graphs namely ladder, circular ladder and Mobius ladders. We compute the eccentricity based topological indices of these graphs and their line graphs. Also, we study the relation between the indices of these graphs with their line graphs. Furthermore, we construct a square grid from the ladder graph and study the eccentricity based topological indices for this grid graph and its line graph.

The Euler Sombor (EU ) index of a graph G is defined aswhere dG(x) and dG(y) denote the degrees of vertex x and y in G, respectively.Biswaranjan Khanra, Shibsankar Das [Euler Sombor index of trees, unicyclic and chemical graphs, MATCH Commun.Math.Comput.Chem.94 (2025) 525-548], posed an open problem about determining the extremal values and extremal graphs for

A molecular graph is a finite simple graph representing the carbon-atom skeleton of an organic molecule of a hydrocarbon. Studying molecular graphs is a constant focus in chemical graph theory: an effort to better understand molecular structure. Many types of graph indices such as degree-based graph indices, distance-based graph indices, and counting-related graph indices have been explored recently. Among degree-based graph indices, Zagreb indices are the oldest and studied well. In the last few years, many new graph indices were proposed. The present survey of these graph indices outlines their mathematical properties and also provides an exhaustive bibliography.

Chemical graph theory is related to the structure of different chemical compounds. A chemical graph represents the molecule of the substance. Chemical graph theory provides the connection between the real number and the different physical, chemical, and biological properties of the chemical species. By implementing the mathematical tools, a chemical graph is converted into a real number. This number can have the predicating ability about the properties of the molecule. In this article, we find some topological indices via M-polynomial for the Starphene graph.

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 and molecular graphs have fundamental applications in chemoinformatics, quantitative structureproperty relationships (QSPR), quantitative structure-activity relationships (QSAR), virtual screening of chemical libraries, and computational drug design. Chemoinformatics applications of graphs include chemical structure representation and coding, database search and retrieval, and physicochemical property prediction. QSPR, QSAR and virtual screening are based on the structure-property principle, which states that the physicochemical and biological properties of chemical compounds can be predicted from their chemical structure. Such structure-property correlations are usually developed from topological indices and fingerprints computed from the molecular graph and from molecular descriptors computed from the three-dimensional chemical structure. We present here a selection of the most important graph descriptors and topological indices, including molecular matrices, graph spectra, spectral moments, graph polynomials, and vertex topological indices. These graph descriptors are used to define several topological indices based on molecular connectivity, graph distance, reciprocal distance, distance-degree, distance-valency, spectra, polynomials, and information theory concepts. The molecular descriptors and topological indices can be developed with a more general approach, based on molecular graph operators, which define a family of graph indices related by a common formula. Graph descriptors and topological indices for molecules containing heteroatoms and multiple bonds are computed with weighting schemes based on atomic properties, such as the atomic number, covalent radius, or electronegativity. The correlation in QSPR and QSAR models can be improved by optimizing some parameters in the formula of topological indices, as demonstrated for structural descriptors based on atomic connectivity and graph distance.

Drugs and chemical compounds are often modeled as graphs in which the each vertex of the graph expresses an atom of molecule and covalent bounds between atoms are represented by the edges between their corresponding vertices. The topological indicators defined over this molecular graph have been shown to be strongly correlated to various chemical properties of the compounds. In this article, by means of graph structure analysis, we determine several distance based topological indices of friendship graph F_{3}^{(n)} which is widely appeared in various classes of new nanomaterials, drugs and chemical compounds.

Chemical compounds and drugs are often modeled as graphs where each vertex represents an atom of molecule and covalent bounds between atoms are represented by edges between the corresponding vertices. This graph derived from a chemical compounds is often called its molecular graph and can be different structures. In this paper, we determine the accurate formulas to compute the generalized Harary index and eccentric connectivity polynomial of certain special molecular graphs.

Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. We call such a graph, which is derived from a chemical compound, a molecular graph. Evidence shows that the vertex-weighted Wiener number, which is defined over this molecular graph, is strongly correlated to both the melting point and boiling point of the compounds. In this paper, we report the extremal vertex-weighted Wiener number of bicyclic molecular graph in terms of molecular structural analysis and graph transformations. The promising prospects of the application for the chemical and pharmacy engineering are illustrated by theoretical results achieved in this paper.