Inequalities between degree- and distance-based graph invariants

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.

In this paper, we introduce a new graph operation based on a central graph called central vertex-edge join (denoted by $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$) and then determine the distance spectrum of $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$ in terms of the adjacency spectra of regular graphs $G_1$, $G_2$ and $G_3$ when $G_1$ is triangle-free. As a consequence of this result, we construct new families of non-D-cospectral D-equienergetic graphs. Moreover, we determine bounds for the distance spectral radius and distance energy of the central vertex-edge join of three regular graphs. In addition, we provide its results related to graph invariants like eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic index, eccentric atom-bond connectivity index, Wiener index. Using these results, we calculate the topological indices of the organic compounds Methylcyclobutane $(C_5H_{10})$ and Spirohexane $(C_6H_{10})$.

The augmented eccentric connectivity index of a graph which is a generalization of eccentric connectivity index is defined as the summation of the quotients of the product of adjacent vertex degrees and eccentricity of the concerned vertex of a graph. In this paper we established some relationships between augmented eccentric connectivity index and several other graph invariants like number of vertices, number of edges, maximum vertex degree, minimum vertex degree, the total eccentricity index the first Zagreb indices and the second multiplicative Zagreb index.

The eccentricity ε u of vertex u in a connected graph G , is the distance between u and a vertex farthermost from u . The aim of the present paper is to introduce new eccentricity based index and eccentricity based polynomial, namely modified augmented eccentric connectivity index and modified augmented eccentric connectivity polynomial respectively. As an application we compute these new indices for octagonal grid O n m $\begin{array}{} \displaystyle O_n^m \end{array}$ and we compare the results obtained with the ones obtained by other indices like Ediz eccentric connectivity index, modified eccentric connectivity index and modified eccentric connectivity polynomial ECP ( G , x ).

A topological index is a numeric quantity derived from the structure of a graph which is invariant under automorphisms of the graph under consideration. In this chapter, the Wiener, Szeged, and Cluj-Ilmenau indices and one-alpha descriptor will be calculated for an infinite family of nanocones, CNC 4[n], and eccentric connectivity; augmented eccentric connectivity; and Wiener, Szeged, PI, vertex PI, and the first and second Zagreb indices of N-branched phenylacetylenes nanostar dendrimers will be obtained. For obtaining Wiener and Szeged indices, we use a powerful method given by Klavžar.

Topological models for prediction of antihypertensive activity of substituted benzylimidazoles

The eccentricity of a vertex is the maximum distance from it to any other vertex, ec(u) = max{d(u, v)|uV(G)}.In this paper we compute total eccentricity index, average eccentricity index, different versions of eccentricity-based Zagreb indices( * (), * * (), * ()), fourth GA eccentricity index, fifth ABC eccentricity index, eccentric connectivity index and eccentric connectivity polynomial of line graphs of CNC 3 [2] and CNC 4 [2] carbon nanocones.

The connective eccentricity index (CEI) of a hypergraph G is defined as ξce(G)=∑v∈V(G)dG(v)εG(v), where εG(v) and dG(v) denote the eccentricity and the degree of the vertex v, respectively. In this paper, we determine the maximal and minimal values of the connective eccentricity index among all k-uniform hypertrees on n vertices and characterize the corresponding extremal hypertrees. Finally, we establish some relationships between the connective eccentricity index and the eccentric connectivity index of hypergraphs.

The augmented eccentric connectivity index is defined as the summation of the quotients of the product of adjacent vertex degrees and eccentricity of the concerned vertex of a graph which is a generalization of eccentric connectivity index. In this paper we present explicit expressions for the values of augmented eccentric connectivity indices of some particular thorn graphs like thorn path, thorn cycle and thorn star and then consider a particular thorn star, dendrimers.

Four highly discriminating fourth-generation topological indices (TIs), termed as superaugmented eccentric distance sum connectivity indices, as well as their topochemical versions (denoted by , , and ), have been conceptualized in this study. The values of these indices for all possible structures with three, four, and five vertices containing one heteroatom were computed using an in-house computer program. The proposed superaugmented eccentric distance sum connectivity topochemical indices exhibited exceptionally high discriminating power, low degeneracy, and high sensitivity toward both the presence and the relative position of heteroatom(s) for all possible structures with five vertices containing at least one heteroatom. Intercorrelation analysis revealed the absence of correlation of proposed indices with Zagreb indices and the molecular connectivity index. Subsequently, the proposed TIs were successfully utilized for the development of models for the prediction of checkpoint kinase inhibitory activity of 2-arylbenzimidazoles. A data set comprising 47 differently substituted analogs of 2-arylbenzimidazoles was selected for the study. The values of various TIs for each analog in the data set were computed using an in-house computer program. The resulting data were analyzed, and suitable models were developed through decision tree (DT), random forest (RF), and moving average analysis (MAA). The performance of the models was assessed by calculating the specificity, sensitivity, overall accuracy, and Mathew's correlation coefficient. A decision tree was constructed for the checkpoint kinase inhibitory activity to determine the importance of topological indices. The decision tree identified the proposed TIs -, - as the most important indices. The decision tree learned the information from the input data with an accuracy of 96% and correctly predicted the cross-validated (10-fold) data with an accuracy of 77%. Random forest correctly predicted the checkpoint kinase inhibitory activity with an accuracy of 83%. The single index-based models were also developed for the prediction of checkpoint kinase inhibitory activity using MAA. The accuracy of prediction of single index-based models derived through MAA was found to vary from a minimum of 90% to a maximum of 95%. Exceptionally high discriminating power, low degeneracy, and high sensitivity toward branching and presence of heteroatom of proposed indices can be of immense use in drug design, isomer discrimination, similarity/dissimilarity studies, quantitative structure activity/property relationships, lead optimization, and combinatorial library design.

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.

Nanostructures are objects whose sizes vary between microscopic and molecular. The most significant of these new elements are carbon nanotubes. These elements have extraordinary microelectronic properties and many other exclusive physiognomies. Recently, researchers have given attention to the mathematical properties of these materials. The aim and objective of this research article is to investigate the most important molecular descriptors namely Wiener, edge-Wiener, vertex-edge-Wiener, vertex-Szeged, edge-Szeged, edge-vertex-Szeged, total-Szeged, PI, Schultz, Gutman, Mostar, edge-Mostar, and total-Mostar indices of three-layered single-walled titania nanosheets. By computing these topological indices, material science researchers can have a better understanding of structural and physical properties of titania nanosheets, thereby synthesizing more easily new variants of titania nanosheets with more amenable physicochemical properties. The cut method turned out to be extremely handy when dealing with distance-based graph invariants which are in turn among the central concepts of chemical graph theory. In this method, we use the Djokovi-Winkler relation to find the suitable edge cuts to leave the graph into exactly two components. Based on the graph theoretical measures of the components, we obtain the desired topological indices by mathematical computations. In this paper, distance-based indices for three-layered single-walled titania nanosheets were investigated and given the exact expressions for various dimensions of three-layered singlewalled titania nanosheets. These indices may be useful in synthesizing new variants of titania nanosheets and the computed topological indices play an important role in studies of Quantitative structure-activity relationship (QSAR) and Quantitative structure-property relationship (QSPR). In this paper, we have obtained the closed expressions of several distance-based topological indices of three-layered single-walled titania nanosheet TNS3 [m, n] molecular graph for the cases m > n and m < n. The graphical validations for the computed indices are done and we observe that the Wiener types, Schultz and Gutman indices perform in a similar way whereas PI and Mostar type indices perform in the same way.

Relationships between Wiener’s index W (a distance based topological descriptor), the Zagreb group parameter M 1 (an adjacency based topological descriptor), and the eccentric connectivity index ξc (an adjacency-cum-distance based topological descriptor) of thiophenes, on the one hand, and their agonist allosteric enhancer activity with respect to human A1 adenosine receptors, on the other hand, have been studied. A training set comprising 59 analogs of substituted thiophenes was selected and the corresponding values of Wiener’s index, the Zagreb group parameter, and the eccentric connectivity index for each compound were calculated. The results were analyzed and suitable models were developed after determination of the activity ranges. Subsequently, the biological activity was assigned using these models to each compound involved in the data set, and the results were compared to the reported agonist allosteric enhancer activity. The overall accuracy of prediction was found to vary from a minimum of 84% for a model based on the eccentric connectivity index to a maximum of 91% for a model based on the Zagreb group parameter.

Let G=(V, E) be a graph, where V(G) is a non-empty set of vertices and E(G) is a set of edges.We defined dv denote the degree of vertex v∈V(G). The Eccentric Connectivity index ξ(G) and theConnective Eccentric index Cξ(G) of graph G are defined as ξ(G)= ∑ v∈V(G)dv x ξ(v) and Cξ(G)=ξ(G)= ∑ v∈V(G)dv x ξ(v)- where ε(v) is defined as the length of a maximal path connecting a vertex v toanother vertex of G. In this present paper, we compute these Eccentric indices for an infinite family oflinear polycene parallelogram benzenod by a new method.Keywords: Molecular graphs; Linear polycene parallelogram; Benzenoid; Eccentric connectivityindex; Connective eccentric index

A topological index is a numerical descriptor of the molecular structure based on certain topological features of the corresponding molecular graph. Topological indices are scientific contemplations of a graph that outline its subatomic topology and are graph-invariant. In a QSAR/QSPR study, topological indices are utilized to anticipate the physico-concoction resources and bioactivity of compounds. In this paper, we study some distance-based topological indices such as eccentric connectivity index (ECI), total eccentricity index (TEI), and eccentricity-based Zagreb index for dominating David-derived networks (DD network) and provide exact formulae of the said indices. These outcomes are valuable to organize the science of hidden topologies of this network.