Sum-connectivity Index Research Articles

On degree based topological indices of self‐assembled tetraphenylethylene and terpyridine rosettes

AbstractA technique is described to compute topological indices for supramolecular rosettes of tetraphenylethylene (TPE) and terpyridine (TPY) ligands with its applications on physicochemical and biological properties. This technique, we have applied to a self‐assembled TPE and TPY supramolecular graphs which is obtained in the form of rosette cycles. Also these type of rosettes graph finds significant applications in electrical sensors, light emitting diodes, bioimaging and photoelectric devices, and so forth. As regarded to the next generation sensing applications with a typical induced aggregative emission behavior, tetraphenylethylene can be utilized in constructing fluorescent probes. For this supramolecular systems we have done a work by computing some topological indices such as the variants of Zagreb Index, Randić Index, Sum Connectivity Index, ABC Index, and Harmonic Index.

Certain topological indices of Basava wheel windmill graph

In this paper, we have proposed new windmill graph, that is Basava wheel windmill graph. The Basava wheel windmill graph \(W^{(m)}_{n+1}\) is the graph obtained by taking \(m\geq 2\) copies of the graph \(K_1+W_{n}\) for \(n\geq 4\) with a vertex \(K_1\) in common. Inspired by recent work on topological indices, proposed new degree-based topological indices namely, general \(SK_{\alpha}\) and \(SK^{\alpha}_1\) indices of a graph \(G\). We have obtained first and second Zagreb index, F-index, first and second hyper-Zagreb index, harmonic index, Randi\(\acute{c}\) index, general Randi\(\acute{c}\) index, sum connectivity index, general sum connectivity index, atom-bond connectivity index, geometric-arithmetic index, Symmetric division deg index, Sombor index, SK indices, general \(SK_{\alpha}\) and \(SK^{\alpha}_1\) indices of Basava wheel windmill graph. Further, we have computed exact values of these topological indices of chloroquine, hydroxychloroquine and remdesiver.

Topological Attributes of Silicon Carbide Si C 4 - II i , j Based on Ve-Degree and Ev-Degree

The study of graphs containing edges and nodes is of great importance in science and in real life. Topological indices TIs are numerical parameters linked to chemical graphs used to estimate the biological, toxicological, and physical properties of chemical compounds. Recently, two terms, ev-degree- and ve-degree-based TIs, have been defined in chemical graph theory. In this paper, we have estimated degree-based TIs, namely, ev-degree Zagreb M type index, ev-degree Randic R index, ve-degree atom-bond connectivity ABC index, ve-degree geometric arithmetic index GA , ve-degree harmonic H index, and ve-degree sum-connectivity index for Si C 4 - II i , j .

On General Sum-Connectivity Index of Trees of Fixed Maximum Degree and Order

On General Sum-Connectivity Index of Trees of Fixed Maximum Degree and Order

On Bond Incident Degree Indices of Connected Graphs with Fixed Order and Number of Pendent Vertices

The graphs having the maximum value of certain bond incident degree indices (including the second Zagreb index, general sum-connectivity index, and general zeroth-order Randić index) in the class of all connected graphs with fixed order and number of pendent vertices are characterized in this paper. The problem of finding graphs having the minimum values of the second Zagreb index and general zeroth-order Randić index from the aforementioned class of connected graphs is also addressed. One of the obtained results about the first Zagreb index has already been proved in the papers [I. Gutman, M. Kamran Jamil, N. Akhter, Trans. Combin. 4 (2015) 43-48] and [M. Enteshari, B. Taeri, MATCH Commun. Math. Comput. Chem. 86 (2021) 17-28]; however, it is proved here by another method with a short proof. Moreover, one of the obtained results concerning the second Zagreb index gives a partial solution to a problem attacked in the aforementiond paper of Enteshari and Taeri.

Extremum sum-connectivity index of trees and unicyclic graphs

The sum-connectivity index of a graph [Formula: see text] is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. In this paper, we provide a simpler method to the extremal trees and unicyclic graph of the sum-connectivity index.

Topological Properties of Concealed Non-Kekulean Benzenoid Hydrocarbon

A topological descriptor is regarded as a numerical parameter derived by using mathematical tools from the molecular graph of different chemical structures. The chemical graph theory is a paramount section of mathematical chemistry. In this section, there are many topological parameters, that have very useful characteristics to study the molecular structure of chemicals. In this article, we investigate topological indices of concealed non-kekulean benzenoid hydrocarbon graph. We will compute general Randic index, general sum connectivity index, sum connectivity index, general version of harmonic index, harmonic index, first, second and third Zagreb indices, SK, SK 1, SK 2 indices, augmented Zagreb index, atomic bond connectivity index, geometric arithmetic index, first and second gourava indices, first and second hyper-gourava indices and 4th version of atomic bond connectivity, 5th version of geometric arithmetic index, and the Sanskurti index.

Topological Invariants for the General Structure of Grape Seed Proanthocyanidins

In this paper, we formulate the degree-based topological indices such as first Zagreb index, second Zagreb index, third Zagreb index, atom bond connectivity index, geometric-arithmetic (GA) index, general sum connectivity index, hyper-Zagreb index, augmented Zagreb index, sum connectivity index and general Randic index Rλ (℘), for λ = © −1, 1, − 1 2 , 1 2 ª for grape seed Proanthocyanidins network. By using above-mentioned topological indices, this study provides us different outcomes.

Multiplicative Attributes Derived from Graph Invariants for Saztec4 Diamond

This study consists of developing some closed and updated formulas derived from multiplicative graph invariants such as general Randic index for λ0 = {±1, ± 1/2}, ordinary general geometric‐arithmetic (OGA), general version of harmonic index (GHI), sum connectivity index (SI), general sum connectivity index (GSI), 1st and 2nd Gourava and hyper‐Gourava indices, (ABC) index, Shegehalli and Kanabur indices, 1st generalised version of Zagreb index (GZI), and forgotten index (FI) for the subdivided Aztec diamond network. Aztec diamond is constructed based on the squares boxes. These square boxes are placed at the centre point and nourish the condition |s − (1/2)| + |r − (1/2)| ≤ n. Furthermore, we put a new vertex of degree‐2 at each edge of the small boxes, squares in shapes. A new structure is obtained that has the same properties as its parental graph and is called a subdivided Aztec diamond and symbolised as Saztecn. Subsequently, we compute the multiplicative topological attributes to get some new formulas. For this purpose, a simple, connected, and the finite graph is considered by supposing it Y as the graph of the Saztecn. The order and size have also been discussed in this study and found three different kinds of edges (2, 2), (2, 3), and (2, 4) for computing. The discussion on the networks mentioned above provides us with essential results that can be used in the determination of bio and physio activities and can be interspersed with the molecular compounds and their graphical structures better to understand their physical as well as biological properties.

Topological Indices of Families of Bistar and Corona Product of Graphs

Topological indices are graph invariants that are used to correlate the physicochemical properties of a chemical compound with its (molecular) graph. In this study, we study certain degree‐based topological indices such as Randić index, Zagreb indices, multiplicative Zagreb indices, Narumi–Katayama index, atom‐bond connectivity index, augmented Zagreb index, geometric‐arithmetic index, harmonic index, and sum‐connectivity index for the bistar graphs and the corona product , where represents the complement of complete graph Kn.

On Vertex-Edge-Degree Topological Descriptors for Certain Crystal Networks

Due to the combinatorial nature of graphs they are used easily in pure sciences and social sciences. The dynamical arrangement of vertices and their associated edges make them flexible (like liquid) to attain the shape of any physical structure or phenomenon easily. In the field of ICT they are used to reflect distributed component and communication among them. Mathematical chemistry is another interesting domain of applied mathematics that endeavors to display the structure of compounds that are formed in result of chemical reactions. This area attracts the researchers due to its applications in theoretical and organic chemistry. It also inspires the mathematicians due to involvement of mathematical structures. Regular or irregular bonding ability of molecules and their formation of chemical compounds can be analyzed using atomic valences (vertex degrees). Pictorial representation of these compounds helps in identifying their properties by computing different graph invariants that is really considered as an application of graph theory. This paper reflects the work on topological indices such as -degree Zagreb index, the first -degree Zagreb index, the first -degree Zagreb index, the second -degree Zagreb index, -degree Randic index, the -degree Randic index, the -degree atom-bond connectivity index, the -degree geometric-arithmetic index, the -degree harmonic index and the -degree sum-connectivity index for crystal structural networks namely, bismuth tri-iodide and lead chloride. In this article we have determine the exact values of -degree and -degree based topological descriptors for crystal networks.

Vertex-Edge Degree Based Indices of Honey Comb Derived Network

Chemical graph theory is a branch of mathematics which combines graph theory and chemistry. Chemical reaction network theory is a territory of applied mathematics that endeavors to display the conduct of genuine compound frameworks. It pulled the research community due to its applications in theoretical and organic chemistry since 1960. Additionally, it also increases the interest the mathematicians due to the interesting mathematical structures and problems are involved. The structure of an interconnection network can be represented by a graph. In the network, vertices represent the processor nodes and edges represent the links between the processor nodes. Graph invariants play a vital feature in graph theory and distinguish the structural properties of graphs and networks. In this paper, we determined the newly introduced topological indices namely, first -degree Zagreb index, first -degree Zagreb index, second -degree Zagreb index, -degree Randic index, -degree atom-bond connectivity index, -degree geometric-arithmetic index, -degree harmonic index and -degree sum-connectivity index for honey comb derived network. In the analysis of the quantitative structure property relationships (QSPRs) and the quantitative structureactivity relationships (QSARs), graph invariants are important tools to approximate and predicate the properties of the biological and chemical compounds. Also, we give the numerical and graphical representation of our outcomes.

Quantitative structure–property relationship of Ev‐degree and Ve‐degree based topological indices : physico‐chemical properties of benzene derivatives

AbstractA quantitative structure–property relationship study was performed for the prediction of physical properties of 22 benzene derivatives using ev‐degree and ve‐degree topological indices. These topological indices correlate certain physico‐chemical property with the chemical structure (like molecular weight, enthalpy, boiling point and π‐electron energy of β‐unit of benzene derivatives). At first, a simple linear regression model was developed using ve‐degree and ev‐degree topological indices and the certain physical properties of the considered benzene derivatives. It is found that the atom bond connectivity index () possesses the best correlating ability among several topological indices to predict the boiling point, the ve‐degree based Randić index is the best predictor of enthalpy, the ev degree based Randić index is the best predictor of π‐electron energy, the sum connectivity index is the best predictor of molecular weight. Second, multiple linear regressions were used for prediction the physical properties (molecular weight, boiling point, enthalpy and π‐electron energy of β‐unit) on the basis of six predictor such as , , , , , and indices. This multi‐linear regression model shows the 100% variation in π‐electron energy of β‐unit. A Maple software based algorithm is used for the calculations of topological indices and the statistical analysis is performed with the SPSS software. The calculations of the ev and ve degree based topological indices of chemical structure of styrene‐butadiene rubber are presented at the end to illustrate the given Maple algorithm.

On Some Properties of Multiplicative Topological Indices in Silicon-Carbon

The use of graph theory can be visualized in nanochemistry, computer networks, Google maps, and molecular graph which are common areas to elaborate application of this subject. In nanochemistry, a numeric number (topological index) is used to estimate the biological, physical, and structural properties of chemical compounds that are associated with the chemical graph. In this paper, we compute the first and second multiplicative Zagreb indices ( M 1 G and ( M 1 G )), generalized multiplicative geometric arithmetic index ( GA α II G ), and multiplicative sum connectivity and multiplicative product connectivity indices ( SCII G and PCII G ) of SiC 4 − I m , n and SiC 4 − II m , n .

Degree-based topological indices of product graphs

In this paper, we obtain the quantitative calculation formula of the degree-based topological indices of four standard products for the path and regular graphs, which unify to solve the question on the product of these basic graphs without dealing with it one by one separately. As applications, we give the corresponding calculation formula of the general Randić index, the first general Zagreb index, and the general sum-connectivity index.

Some new degree based topological indices of h-naphtalenic graph via M-polynomial approach

In this object, we present some new formulas of the reduced reciprocal Randic index, the arithmetic geometric 1 index, the SK, SK1, SK2 indices, first Zagreb index, the general sum-connectivity index, the SCI index and the forgotten index. They were utilized for new degree-based topological indices via M-polynomial. We retrieved these topological indices for H-Naphtalenic nanotubes.

Properties of Total Transformation Graphs for General Sum-Connectivity Index

The study of networks and graphs through structural properties is a massive area of research with developing significance. One of the methods used in studying structural properties is obtaining quantitative measures that encode structural data of the whole network by the real number. A large collection of numerical descriptors and associated graphs have been used to examine the whole structure of networks. In these analyses, degree-related topological indices have a significant position in theoretical chemistry and nanotechnology. Thus, the computation of degree-related indices is one of the successful topics of research. The general sum-connectivity GSC index of graph Q is described as χ ρ Q = ∑ qq ′ ∈ E Q d q + d q ′ ρ , where d q presents the degree of the vertex q in Q and ρ is a real number. The total graph T Q is a graph whose vertex set is V Q ∪ E Q , and two vertices are linked in T Q if and only if they are either adjacent or incident in Q . In this article, we study the general sum-connectivity index χ ρ Q of total graphs for different values of ρ by using Jensen’s inequality.

On the Vertex-Degree Based Invariants of Digraphs

Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $\varphi$ of $D$ is defined as a summation over all arcs, $I(D) = \frac{1}{2}\sum_{uv\in A}{\varphi(d_u^+,d_v^-)}$, where $d_u^+$ (resp. $d_u^-$) denotes the out-degree (resp. in-degree) of a vertex $u$. In this paper, we give the extremal values and extremal digraphs of $I(D)$ over all digraphs with $n$ non-isolated vertices. Applying these results, we obtain the extremal values of some vertex-degree based topological indices of digraphs, such as the Randi\'{c} index, the Zagreb index, the sum-connectivity index, the $GA$ index, the $ABC$ index and the harmonic index, and the corresponding extremal digraphs.

Correlation Ability of Degree-Based Topological Indices for Physicochemical Properties of Polycyclic Aromatic Hydrocarbons with Applications

Several chemical and medical experimentation reveals a dependence of physicochemical and biological properties of a compound on its molecular structure. Molecular/topological descriptors/indices retrieve this dependence by employing mathematical/statistical tools to generate quantitative structure property/activity relationship (QSAR/QSPR) models. QSAR/QSPR models are regression models which theoretically relate a physicochemical/biological property to a molecular descriptor. By converting a compound to a chemical graph, graph-theoretic topological indices use algorithmic graph theory to generate QSAR/QSPR models. Degree-related molecular indices have been well-known for correlating better with certain physicochemical characteristics of compounds. In this paper, we conduct a comparative analysis to put forward the best degree-based index for correlating with physicochemical characteristics of polycyclic aromatic hydrocarbons. Two different computational techniques have been put forward and then used to conduct our comparative analysis. Our comparative testing generates several favorable and unexpected conclusions as the well-known indices such as the augmented Zagreb index, the atom-bond connectivity index, the geometric arithmetic index deliver relatively poor performance. Moreover, exceptional performance of the generalized variants of Randić and sum-connectivity indices is somehow unexpected as they significantly outperform the well-studied and well-researched degree-based indices. This warrants to study further the general Randić and sum-connectivity indices for QSAR/QSPR models. Applications to certain families of carbon polyhex aromatic nanotubes are reported.

Sharp Lower Bounds of the Sum-Connectivity Index of Unicyclic Graphs

The sum-connectivity index of a graph G is defined as the sum of weights 1 / d u + d v over all edges u v of G , where d u and d v are the degrees of the vertices u and v in graph G , respectively. In this paper, we give a sharp lower bound on the sum-connectivity index unicyclic graphs of order n ≥ 7 and diameter D G ≥ 5 .