Document Type : Original Research Article
Authors
- Ashwini Ankanahalli Shashidhara ^{} ^{1}
- Hanan Ahmed ^{} ^{} ^{2}
- Murat Cancan ^{} ^{3}
^{1} Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru-570 006, India
^{2} Department of Mathematics, Ibb University, Ibb, Yemen
^{3} Faculty of Education, Van Yuzuncu Yıl University, Van, Turkey
Abstract
In graph theory, topological indices and domination parameters are essential topics. A dominating set for a graph G=(V(G),E(G)) is a subset D of V(G) such that every vertex not in D is adjacent to at least one vertex of D introduced novel topological indices known as domination topological indices. In this research work, we found exact values to determine Sombor index of some families of graphs including the join and corona product. Some bounds for these new topological indices were also found. Likewise, we defined the significance of the Sombor index in predicting the physicochemical properties of butane derivatives.
Graphical Abstract
Keywords
Introduction
Mathematics, chemical graph theory is one of the important branch which is combines with graph theory and chemistry. Graph theory is used to mathematically model molecules to gain insight into the physical properties such as boiling point, melting point, density, and many more chemical compounds, are related to the geometric structure of the compound. Chemical graph theory, which deals with the non-trivial applications of graph theory to solve molecular problems [7]. Topological indices are the most significant molecular descriptor, and it is a mathematical formula that can be applied to any graphs which are related to some chemical components. Topological indices have very important and huge applications (for more details, [6,8,16,21-25,27-38]). Molecular descriptors play a significant role in chemical graph theory. Now a days, there are numerous topological indices some of which are applied to the physiochemical properties of chemical compounds.
Consider G is a graph with no loops and no multiple edges. In G, V(G) is the vertex set and E(G) is the edge set. The vertex of the graph corresponds to an atom, while the edge represents the chemical bond of molecules. The vertex v in G of order n is said to be full degree if d(v)=n-1. A vertex subset D is said to be dominating set such that every vertex v(vɆD) is adjacent to at least one vertex of D. If D-v(vɛD) is not a dominating set, then it is called a minimal dominating set and the cardinality of minimum dominating set is called domination number, denoted by y(G) [12]. For more details about domination in the graph, [13,14,18,20]. Hanan Ahmed et al. [11] introduced a new degree based on a minimal dominating set, denoted by d_{d}(v), which is defined as the number of minimal dominating set containing v. We use the notation T_{m}(G) which indicates the total number of minimal dominating set of G. For more details regarding domination topological indices of some families of standard graph and graph operations. we refer to study [11]. Likewise, for applications of domination topological indices readers can refer [2,3,4].
Definition 1.1 [4] Let G be a simple, connected graph. Then,
- The first, the second domination Zagreb, and modified first domination Zagreb indices are defined by:
- The forgotten domination, hyper domination, and modified forgotten domination indices of graphs are defined as follow:
Definition 1.2 [18] Sombor index of graph G is one of the topological index which is defined as:
In this paper, we introduced new topological indices defined on domination degree named as domination Sombor index defined as:
Domination Sombor index of graphs
Definition 2.1 Let G be a simple connected graph. Then, the domination Sombor index is defined as:
The Windmill graph Wd^{s}_{r }is an undirected graph constructed for r≥2 and s≥2 by s copies of the complete graph k_{r} at a shared universal vertex [5].
The significance of the Sombor index in predicting the physic-chemical properties of butane derivatives
Quantitative structure-property relationships (QSPR) remain the focus of many studies aimed at the modeling and prediction of physicochemical and biological properties of molecules. A powerful tool to help in this task is chemometrics, which uses statistical and mathematical methods to extract maximum information from a data set. QSPR uses chemometric methods to describe how a given physicochemical property varies as a function of molecular descriptors describing the chemical structure of the molecule. Thus, it is possible to replace costly biological tests or experiments of a given physicochemical property with calculated descriptors, which can, in turn, be used to predict the responses of interest for new compounds. The basic strategy of QSPR is to find an optimum quantitative relationship, which can be used for the prediction of the properties of compounds, including those unmeasured. It is obvious that the performance of QSPR model mostly depends on the parameters used to describe the molecular structure. Many efforts have been made to develop alternative molecular descriptors which can be derived by using only the information encoded in the chemical structure. Much attention has been concentrated on “topological indices” derived from the connectivity and composition of a molecule which has made significant contributions in QSPR studies. The topological index has advantages of simplicity and quick speed of computation and thus, is of high singificance for the scientists.
In this section, we are going to discuss the QSPR analysis of domination topological indices.
Furthermore, we show that the characteristics have a good correlation with the physicochemical characteristics of butane derivatives, as represented in Table 1. The exact values of domination topological indices of butane derivatives are listed in Table 2.
The correlation coefficient values of predicted physiochemical properties of butane derivative with the exact values of this physico-chemical properties are given in Tables 6 (a), (b) and 7.
The following figures indicates how much the predicted values of physio-chemical properties are correlated with the well-known physio-chemical properties. The degree of correlation between any two data sets is measured by the correlation coefficent (R). During the R value becomes closed to unity, two datasets are more correlated. The QSPR study of domination topological indices reveals that these domination indices can be helpful in predicting Surface Tension (ST), complexity, Heavy Atomic Count (H.A.C), density, and refraction index. We can also note that the correlation coefficient values of the predicted values of Surface Tension (ST) and its exact values lies between the range of 0.53≤R≤0.83 with the best correlation coefficient 0.83 of domination Sombor index (Figure 1). Furthermore, we can see the correlation coefficient values of the complexity predicted values with its exact values are lies between the range of 0.53≤R≤0.90. Here, also the best value of correlation coefficient is 0.904 for domination Somber index (Figure 2). While based on Figure 3, the correlation range is 0.58≤R≤0.88 which shows a good correlation of predicted values of (H.A.C) with exact values of (H.A.C). Figure 4 demonstrates that the correlation coefficient of density predicted values with its exact values are lies between the range of 0.81≤R≤0.77. Finally, as Figure 5 indicates, the correlation range is 0.654≤R≤0.76 which shows a good correlation coefficient of predicted values of refraction index with its exact values. In Figure 5, we can see that the correlation coefficient values of the domination Sombor index are the most highest. On the another hand, all domination indices are very useful to predict the physio-chemical properties of butane derivatives. Surface Tension (ST), complexity, Heavy Atomic Count (H.A.C), density and refraction index are important physico-chemical properties by using those domination indices to predict the values of these properties. It has been shown that these indices can beconsidered useful molecular descriptors in QSPR analysis of butane derivative.
Conclusion
In this research work, we defined a new topological index based on the minimal dominating sets. This index is the domination Sombor index. We calculated the exact values of the domination Sombor index of some families and some graph operations. Likewise, we defined the significance of the Sombor index in predicting the physicochemical properties of butane derivatives.
Acknowledgements
Conflict of Interest
Orcid:
Ashwini Ankanahalli Shashidhara: https://orcid.org/0000-0002-7009-321X
Hanan Ahmed: https://orcid.org/0000-0002-4008-4873
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