On topological indices for small RNA graphs

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.

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 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.

Maximum values of Szeged index and edge-Szeged index of graphs

The advent of fuzzy sets, and consequently fuzzy graphs, has solved many problems in ambiguous and uncertain contexts. It is interesting and necessary to study the Wiener index in a cubic fuzzy graph that employs both fuzzy membership and interval-valued fuzzy membership at the same time. In this paper, the Wiener index in a cubic fuzzy graph is introduced as a cubic fuzzy number and some related results are described. The comparison between connectivity index and Wiener index, changes in Wiener index through deleting a node or an edge, and determining the Wiener index in some specific cubic fuzzy graphs have been the other topics studied in this research. In addition, the Wiener index is determined by mentioning concepts of the saturated cubic fuzzy cycle. In this review, the Wiener index is shown as a combination of classical and interval numbers. The results indicate that when some vertices are removed, the Wiener index may change. However, this change will not be exclusively related to both values. Finally, an application of the Wiener index is presented in the study of the properties of some monomer molecules.

A topological index is a number generated from a molecular structure that signifies the molecule’s fundamental structural characteristics. This correlation between the index and various physical attributes is based on an algebraic quantity related to the chemical structure. Many topological indices, such as the Wiener index, first and second Zagreb indices, can be employed to determine various properties, including chemical activity, thermodynamic properties, physicochemical activity, and biological activity. Meanwhile, the p-subgroup graph of a group G is defined as a graph whose vertices represent the elements of the group, and two vertices are adjacent if and only if the order of the subgroup is a prime power. The main objective of this paper is to establish the general formula for certain topological indices, specifically the Wiener index, first Zagreb index, and second Zagreb index for the -subgroup graph associated with dihedral groups.

Topological indices are numerical values that can be analysed to predict the chemical properties of the molecular structure and the topological indices are computed for a graph related to groups. Meanwhile, the conjugacy class graph of is defined as a graph with a vertex set represented by the non-central conjugacy classes of . Two distinct vertices are connected if they have a common prime divisor. The main objective of this article is to find various topological indices including the Wiener index, the first Zagreb index and the second Zagreb index for the conjugacy class graph of dihedral groups of order where the dihedral group is the group of symmetries of regular polygon, which includes rotations and reflections. Many topological indices have been determined for simple and connected graphs in general but not graphs related to groups. In this article, the Wiener index and Zagreb index of conjugacy class graph of dihedral groups are generalized.

An Interactive Framework for RNA Secondary Structure Prediction with a Dynamical Treatment of Constraints

To better elucidate RNA structure-function relationships and to improve the design of pharmaceutical agents that target specific RNA motifs, an understanding of RNA primary, secondary, and tertiary structure is necessary. The prediction of RNA secondary structure from sequence is an intermediate step in predicting RNA three-dimensional structure. RNA secondary structure is typically predicted using a nearest neighbor model based on free energy parameters. The current free energy parameters for 2 × 3 nucleotide loops are based on a 23-member data set of 2 × 3 loops and internal loops of other sizes. A database of representative RNA secondary structures was searched to identify 2 × 3 nucleotide loops that occur in nature. Seventeen of the most frequent 2 × 3 nucleotide loops in this database were studied by optical melting experiments. Fifteen of these loops melted in a two-state manner, and the associated experimental ΔG°(37,2×3) values are, on average, 0.6 and 0.7 kcal/mol different from the values predicted for these internal loops using the predictive models proposed by Lu, Turner, and Mathews [Lu, Z. J., Turner, D. H., and Mathews, D. H. (2006) Nucleic Acids Res. 34, 4912-4924] and Chen and Turner [Chen, G., and Turner, D. H. (2006) Biochemistry 45, 4025-4043], respectively. These new ΔG°(37,2×3) values can be used to update the current algorithms that predict secondary structure from sequence. To improve free energy calculations for duplexes containing 2 × 3 nucleotide loops that still do not have experimentally determined free energy contributions, an updated predictive model was derived. This new model resulted from a linear regression analysis of the data reported here combined with 31 previously studied 2 × 3 nucleotide internal loops. Most of the values for the parameters in this new predictive model are within experimental error of those of the previous models, suggesting that approximations and assumptions associated with the derivation of the previous nearest neighbor parameters were valid. The updated predictive model predicts free energies of 2 × 3 nucleotide internal loops within 0.4 kcal/mol, on average, of the experimental free energy values. Both the experimental values and the updated predictive model can be used to improve secondary structure prediction from sequence.

In chemistry, the molecular structure can be represented as a graph. Based on the information from the graph, its characterization can be determined by computing the topological index. Topological index is a numerical value that can be computed by using some algorithms and properties of the graph. Meanwhile, the non-commuting graph is a graph, in which two distinct vertices are adjacent if and only if they do not commute, where it is made up of the non-central elements in a group as a vertex set. In this paper, the Szeged index of the non-commuting graph of some finite groups are computed. This paper focuses on three finite groups which are the quasidihedral groups, the dihedral groups, and the generalized quaternion groups. The construction of the graph is done by using Maple software. In finding the Szeged index, some of the previous results and properties of the graph for the quasidihedral groups, the dihedral groups, and the generalized quaternion groups are used. The generalisation of the Szeged index of the non-commuting graph is then established for the aforementioned groups. The results are then applied to find the Szeged index of the non-commuting graph of ammonia molecule.

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.

Several bioactivities of chemical compounds in a molecular graph can be expected by using many topological descriptors. A topological descriptor is a numeric quantity which quantify the topology of a graph. By defining the metric on a graph related with a vector space, we consider this graph in the context of few topological descriptors, and quantify the Wiener index, hyper Wiener index, Reciprocal complimentary Wiener index, Schultz molecular topological index and Harary index. We also provide the graphical comparison of our results to describe the relationship and dependence of these descriptors on the involved parameters.

Topological indices are widely used for characterizing molecular graphs, establishing relationships between structure and properties of molecules. In this article we discuss adjacency matrix and three topological indices: Wiener index, Laplacian energy and Zagreb indices of zero-divisor graphs of $$\mathbb {Z}_n$$ . We also provide a MATLAB code for Laplacian energy and Zagreb indices of $$\varGamma (\mathbb {Z}_n)$$ .

A topological index is a numerical value associated with chemical constitution for correlation of chemical structure with various physical properties and chemical reactivity. It is calculated from a graph representing a molecule. Meanwhile, the non-commuting graph, $$\Gamma _G$$ of G, is defined as a graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if and only if they do not commute. The main objective of this article is to determine the general formula of some topological indices, namely Wiener index, first Zagreb index and second Zagreb index for the non-commuting graph associated with generalised quaternion group in terms of n.

Any graph that depicts a particular molecular structure can be given a topological graph index, also known as a molecular descriptor. From this index, it is possible to examine numerical data and learn more about some of the physical characteristics of a molecule. In this study, we construct and explore a novel variation for graphs termed the exponential Wiener index (EW(G)) and the multiplicative exponential Wiener index [Formula: see text]. Typical graphs for which it is also conceivable include the wheel graph, fan graph, star graph, full graph, cycle graph, complete graph, and complete bipartite graph. Additionally, some EW(G) bounds are determined using exponential inequalities. In the end, an algorithm is created to obtain EW(G), and the most effective is investigated. This work is also extended to the molecular graphs for n-alkanes, n = 1,2,..., 10 and benzenoid hydrocarbons. Additionally, a QSPR study that takes into account the molecular mass, melting point, and boiling point of alkanes and total π-electronic energy [Formula: see text], enthalpy of formation [Formula: see text], and boiling points have been conducted. The influential measures such as DFBETAS, DFFITS, Hadi's measure, are calculated by which the influential parameters and the outliers are identified for the regression models between the topological index and these physical parameters.