On Euler-Sombor index of benzenoids and phenylenes

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.

Valence molecular connectivity indices are indices based on the concept of valence delta, d v, that can be derived from general chemical graphs or chemical pseudographs. A general graph or pseudograph is a graph with multiple edges and loops and it can be used to encode, through the valence delta, chemical entities like the sigma-, pi- and non-bonding n-electrons. Two other graph-theoretical concepts that can also be derived from chemical pseudographs are the intrinsic (I) and the electrotopological state (E) values that are the main tools used to define the valence delta of the pseudoconnectivity indices, ?I,S. Complete graphs can, instead, be used to encode, through a new type of valence delta, the core electrons of any type of atoms in a molecule. The connectivity indices either valence connectivity or pseudoconnectivity indices are the starting point to develop, by the aid of a dual procedure, the dual connectivity indices, i.e., the dual connectivity, valence connectivity and pseudoconnectivity indices. The dual indices show the interesting property that not only some of them can assume negative values but also that they can cover a wide range of numerical values. Graph concepts can also be used to deal with the problem of the hydrogen contribution in hydrogen depleted chemical graphs, which are the normal type of graphs used in chemistry. For this purpose a perturbation parameter can be introduced into the definition of the valence delta that allows to differentiates among compounds with similar hydrogen-suppressed chemical graphs but different number of hydrogen atoms, like CH3F and BH2F. The new definition of the central parameter of the molecular connectivity theory, the valence delta, consent to define of a completely new set of connectivity indices, which can be distinguished by their configuration and that can advantageously be used to model different properties and activities of compounds.

In mathematical chemistry, topological indices are molecular descriptors that are calculated on the molecular graph of a chemical compound. The molecular graph is a graph which is obtained from some chemical structures. The degree of every molecular graph cannot exceeds 4. Topological indices are numerical quantities of a graph that describe its topology. An atom represents a vertex and a bond between two atoms represents an edge in a molecular graph. Mainly there are three types of topological indices viz., degree-based, distance based and eigenvalue-based topological indices. The first degree-based topological indices are the first and second Zagreb indices. The first Zagreb index M_1 is defined as the sum of squares of degrees of each vertex in a graph G and the second Zagreb index M_2 is the product of degree of every adjacent vertices. In this case the summation goes on the set of edges of a graph G. The most studied topological indices are degree-based topological indices. Motivated by these topological indices in this paper, we introduce five new degree-based topological indices based on the neighborhood degree of a vertex. Further, we compute the values of various nanostructures like hexagonal parallelogram P(m,n) nanotube, triangular benzenoid G_n,zigzag-edge coronoid fused with starphene nanotubes ZCS(k,l,m), dominating derived networks D_1,D_2,D_3, Porphyrin Dendrimer, Zinc-Porphyrin Dendrimer, Propyl Ether Imine Dendrimer, Poly(Ethylene amido amine Dendrimer, PAMAM dendrimers(????????1,????????2,????????1), linear polyomino chain L_n,Z_n,B_n^1 (n≥3),B_n^2 (n≥3) and triangular, hourglass, and jagged-rectangle benzenoid systems of these indices. The standard computational techniques are used for the computation of topological indices of nanostructures. For the edge partition of the nanostructures the algebraic techniques are used. Using these techniques computation of topological indices became easy and also helped to get the more accurate results.

Topological molecular descriptor is calculated by a mathematical procedure from the structure of chemical compounds represented by a molecular graph, containing information about their structural characteristics [1]. There are many defined topological descriptors in literature which transform specific chemical information into useful numerical values that have been used for correlation of structure-descriptors with various physico-chemical properties [2]. Among the first topological descriptors that accelerate further development of the field are M1 and M2 indices introduced by Gutman and Trinajstić in 1972 [3] (later named Zagreb indices). Vertices of a molecular graph represent atoms and edges connecting vertices of a graph represent carbon-carbon (or non-hydrogen) chemical bonds. The total number of edges connecting one vertex with its (first) neighbour carbon vertices corresponds to valence of that atom with other carbons within a molecule. One of the largest classes of topological descriptors is the one based on the analysis of vertex degrees of a molecular graph [4]. Starting from electronic version of chemical structures we will develop and optimize an application for calculation of different vertex-degree-based topological descriptors, which is made available for free use to the research community at http://meteo2.irb.hr/indexer.

Topological indices are a class of numerical invariants that predict certain physical and chemical properties of molecules. Recently, two novel topological indices, named as Sombor index and reduced Sombor index, were introduced by Gutman, defined as where denotes the degree of vertex in . In this paper, our aim is to order the chemical trees, chemical unicyclic graphs, chemical bicyclic graphs and chemical tricyclic graphs with respect to Sombor index and reduced Sombor index. We determine the first fourteen minimum chemical trees, the first four minimum chemical unicyclic graphs, the first three minimum chemical bicyclic graphs, the first seven minimum chemical tricyclic graphs. At last, we consider the applications of reduced Sombor index to octane isomers.

Graph parameters and topological indices allow a discussion about hierarchical criteria for ranking monocyclic and polycyclic molecular (constitutional) chemical graphs. For brevity, chemical constitutional graphs will be referred to as CGs. These criteria include the number of vertices (graph order), cyclomatic numbers, vertex degrees (from one to four), number of polygons with increasing numbers of edges (3-, 4-, 5-gons, etc.), and vicinity with vertices of increasing rank. A similar hierarchy can be established for ranking vertices in a CG, and a detailed discussion is presented for ordering vertices in all seven identity CGs with 6 vertices and in 25 from the many identity CGs with 7 vertices. This is the first discussion of ranking the vertices of CGs from the viewpoint of centrality measures.

A molecular graph consists of bonds and atoms, where atoms are present as vertices and bonds are present as edges. We can look at topological invariants and topological polynomials that furnish bioactivity and physio-chemical features for such molecular graphs. These topological invariants, which are usually known as graph invariants, are numerical quantities that relate to the topology of a molecular graph. Let m pq ( X ) be the number of edges in X such that ( ζ a , ζ b ) = ( p , q ), where ζ a (or ζ b ) present the degree of a (or b ). The M -polynomial for X can be determined with the help of relation M ( X ; x , y ) = ∑ p ≤ q m p q ( X ) x p y q M(X;x,y)={\sum }_{p\le q}{m}_{pq}(X){x}^{p}{y}^{q} . In this study, we calculate the M -polynomial, forgotten polynomial, sigma polynomial and Sombor polynomial, and different topological invariants of critical importance, referred to as first, second, modified and augmented Zagreb, inverse and general Randić, harmonic, symmetric division; forgotten and inverse invariants of chemical structures namely metal-organic networks (transition metal-tetra cyano benzene organic network) and cuboctahedral bimetallic networks (MOPs) are retrieved using a generic topological polynomial approach. We also draw the two-dimensional graphical representation of outcomes that express the relationship between topological indices and polynomial structural parameters.

Coumarins, a subgroup of colorless and crystalline oxygenated heterocyclic compounds originally discovered in the plant Dipteryx odorata, were the subject of a recent study investigating their quantitative structure-activity relationship (QSAR) in cancer pharmacotherapy. This study utilized graph theoretical molecular descriptors, also known as topological indices, as a numerical representation method for the chemical structures embedded in molecular graphs. These descriptors, derived from molecular graphs, play a pivotal role in quantitative structure-property relationship (QSPR) analysis. In this paper, intercorrelation between the Balban index, connective eccentric index, eccentricity connectivity index, harmonic index, hyper Zagreb index, first path Zagreb index, second path Zagreb index, Randic index, sum connectivity index, graph energy and Laplacian energy is studied on the set of molecular graphs of coumarins. It is found that the pairs of degree-based indices are highly intercorrelated. The use of these molecular descriptors in structure-boiling point modeling was analyzed. Finally, the curve-linear regression between considered molecular descriptors with physicochemical properties of coumarins and coumarin-related compounds is obtained.

A topological index is a real number that is obtained from a chemical graph's structure. Determining the physiochemical and biological characteristics of a variety of medications is useful since it more accurately represents the theoretical characteristics of organic molecules. This is accomplished using degree-based topological indices. The QSPR research has improved the structural understanding of the physiochemical properties of fungicides. Thirteen fungicides are examined for some of their physiochemical properties, and a QSPR model is built using nine of the drugs' topological indices. Here, we examine the degree to which the topological indices and physiochemical attributes are connected. To do this, we create networks connecting each of the topological indices to the properties of fungicides and computationally construct topological indices of the drugs mentioned above. According to this QSPR model, the melting point, boiling point, flash point, complexity, surface tension, etc. of fungicides are strongly connected. It was discovered that the topological indices (TIs) applied to the fungicides more accurately represent their theoretical features and show a strong correlation with their physical attributes.

A graph G consists of a set of vertices V G and a set of edges E G . If the vertices u, v ∈ V G are connected by an edge e then we write e = uv. In chemical graphs, each vertex represents an atom of the molecule, and covalent bonds between atoms are represented by edges between the corresponding vertices. This shape derived from a chemical compound is often called its molecular graph, and can be a path, a tree, or in general a graph. The graph G is said to be connected if for every vertices x and y in V G there exists a path between x and y. The distance dG u v or d u v , between vertices u and v of a connected graph G is the number of edges in a minimum path from u to v. A topological index is a real number related to a molecular graph, which is a graph invariant. There are several topological indices already defined and many of them have found applications as means to model chemical, pharmaceutical and other properties of the molecules. The Wiener index W is the first topological index proposed to be used in Chemistry. It was introduced in 1947 by Harold Wiener,1 as the path number for characterization of alkanes. It is defined as the sum of distances between all pairs of vertices in the graph under consideration. Here, we consider a new topological index, named Padmakar-Ivan index and abbreviated as PI index.2–4 To define the PI index of a connected graph G, we correspond to an edge e = uv of E G , two quantities neu(e G) and nev(e G) in which neu(e G) is the number of edges lying closer to the vertex u than the vertex v, and nev(e G) is the number of edges lying closer to the vertex v than the vertex u. Then the PI index of the graph G is defined as PI G =∑e=uv∈E G [neu(e G +nev(e G)].

The study discussed the QSPR analysis of the mentioned topological indices. The study also demonstrated that the characteristics obtained are highly correlated with those of Hepatocellular Carcinoma (Liver Cancer) drugs through linear regression. drugs are represented as molecular graphs where each vertex represents an atom and each edge represents a link between two atoms. Consider G (V, E) as a molecular graph, where V is the set of vertices and E is the set of edges. In this study, used 6 degree based topological indices, M1(G), M2(G), F(G), S(G), Y(G) and D(G). These indices were used to model five representative physical properties of five liver cancer drugs: BP, FP, P, ST, and MV. The values for these properties were obtained from ChemSpider. The study concluded that degree-based topological indices are effective molecular descriptors for predicting the physical properties of liver cancer drugs. The regression models revealed significant correlations between Surface Tension (ST) and indices such as the Forgotten Index (F(G)) and the Sum-Connectivity Index (S(G)). Although other properties, such as Boiling Point (BP) and Flash Point (FP), demonstrated weaker correlations, the overall findings suggest that topological indices can be valuable tools in Quantitative Structure-Property Relationship (QSPR) studies

Graph theory plays a central role in mathematics, biology, chemistry, computer science, and related disciplines. It has many applications in everyday life, particularly in chemistry, biology, and network theory. Chemical graph theory is a subfield devoted to the mathematical representation and analysis of molecular structures. It is also used in the calculation of topological indices and the prediction of many chemical properties. A topological index is a numerical parameter that characterizes the molecular structure based on its corresponding molecular graph. Consider a simple molecular graph G = (V(G), E(G)) with no loops, multiple edges, or directed edges. Numerous topological indices have been defined and studied for molecular graphs. The vertex and edge eccentric connectivity indices, along with their modified versions, play a significant role in QSPR/QSAR studies within the framework of chemical graph theory. Recently, various studies have been conducted on the backbone DNA graphs. The repeating cycles in the backbone DNA graphs indicate that the graph possesses a periodic and regular symmetry. This symmetry is taken into account in deriving closed formulas for topological index values such as the eccentric connectivity indices. In this paper, some eccentric connectivity indices based on vertices and edges of backbone DNA graphs DNAn have been computed. Furthermore, the two-dimensional plots of DNAn were generated using Cartesian coordinates.

In the study of QSPR/QSAR, topological indices such as Zagreb index, Randic index, atom-bond connectivity index are exploited to estimate the bioactivity of chemical compounds. Inspired by many degree based topological indices, we propose here a new topological index, called the Atom Bond Connectivity temperature index ABCT(G) of a molecular graph G, which shows good correlation with entropy, acentric factor, enthalpy of vaporization and standard enthalpy of vaporization of an octane isomers. In this paper we compute the Atom Bond Connectivity temperature index ABCT(G) of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC_4 C_8 [p,q].

Graphene, composed of a single layer of carbon atoms arranged in a hexagonal lattice pattern, has been the focus of extensive research due to its remarkable properties and practical applications. Topological indices (TIs) play a crucial role in studying graphene's structure as mathematical functions mapping molecular graphs to real numbers, capturing their topological characteristics. To compute these TIs, we employ the M-polynomial approach, an efficient method for deriving degree-based descriptors of molecular graphs. In this study, we analyze the neighborhood multiple M-polynomial of graphene's structure and use it to derive eleven neighborhood multiple degree-based TIs. These TIs allow us to predict various properties of graphene theoretically, bypassing the need for experiments or computer simulations. Furthermore, we showcase various numerical and graphical representations emphasizing the intricate connections between TIs and structural parameters. These computations were further employed to analyze the Quantitative Structure-Property Relationship (QSPR) between TIs and the mechanical properties of graphene, such as Young's Modulus, Poisson's Ratio, Shear Modulus, and Tensile Strength. The results showed strong correlations between neighborhood multiple TIs and Poisson's Ratio and Shear Modulus, underscoring their predictive power for these mechanical properties. These findings highlight the effectiveness of neighborhood multiple degree-based TIs in characterizing and predicting the mechanical properties of graphene structures, providing valuable insights for future applications in material science.

A topological index being a graph theoretic parameter plays a role of function for the assignment of a numerical value to a molecular graph which predicts the several physical and chemical properties of the underlying molecular graph such as heat of evaporation, critical temperature, surface tension, boiling point, octanol-water partition coefficient, density and flash points. For a (molecular) graph Γ, the Lanzhou index (Lz index) is obtained by the sum of deg(v)2de¯g(v) over all the vertices, where deg(v) and de¯g(v) are degrees of the vertex v in Γ and its complement Γ¯ respectively. Let Vαβ be a class of unicyclic graphs (same order and size) such that each graph of this class has order α and β leaves (vertices of degree one). In this note, we compute the lower and upper bounds of Lz index for each unicyclic graph in the class of graphs Vαβ. Moreover, we characterize the extremal graphs with respect to Lz index in the same class of graphs.