Abstract
The topological index is a molecular predictor that is commonly supported in the research of QSAR of pharmaceuticals to numerically quantify their molecular features. Theoretical and statistical study of drug-like compounds improves the drug design and finding work-flow by rationalizing lead detection, instant decision, and mechanism of action comprehension. Using molecular structure characterization and edge segmentation technique, we computed the general temperature topological indices for OTIS networks.
1. Introduction
Mathematical chemistry may be a theoretic science in which artificial structures square measure by the employment of scientific instruments. The artificial diagram hypothesis maybe a part of this field where chart hypothesis devices square measure applied to scientifically demonstrate concoctions. As the graph has vertices that can be pictured by processor nodes and edges represent links between these nodes/processors [1]. The primary application in chemistry was the boiling purpose of the paraffin [1–3].
Topological indices are numerical values that are attributed to a molecular structure. Today, topological indices have been considered by many researchers because they have applications in various sciences, see [4–8].
Aslam et al. [9], obtained new results for OTIS networks by using some of the topological indices. Zahra et al. [3], discussed the swapped networks by using topological indices. In [10], the authors computed some of the topological indices for OTIS networks. Baig et al. [11] discussed some of the DOX and DSL networks by using the topological indices. In [6], the authors obtained some of the new results for anticancer drugs by using the multiplicative topological indices. In [12], the author discussed carbon nanocones and nanotori by using some of the topological indices. Therefore, in this paper, we obtain new results for OTIS networks by using some topological indices.
For a simple graph , we denoted the vertex set by and the edge set by . The degree of the vertex is denoted by .
For any graphs with vertices, the temperature of a vertex is defined in [13] as
Kulli in [14] defined the following topological indices as The general first temperature index The general second temperature index The general temperature indexwhere .
2. Results for Network
In this section, we compute the exact formulas for the OTIS networks of some general temperature indices, and we also examine the relationships between these topologies with graphical diagrams.
Let be the path of vertices and be OTIS (swapped) network with basis network , see Figure 1.
We start by computing the general first temperature index.
Theorem 1. For network, we have
Proof. The OTIS networks have vertices and , edges. Hence, there are three partitions, , and . Hence, we can write thatIt can be easily seen that , and .
By applying the definitions and Equalities (6)–(9), we can write
Now, we compute the general second temperature index.
Theorem 2. For network, we have
Proof. By applying the definitions and equalities (6)–(9), we have
Here, we compute the general temperature index.
Theorem 3. For network, we have
Proof. By applying the definitions and Equalities (6)–(9), we can write
3. Results for Network
The OTIS (swapped) network is derived from the graph , which a graph with vertex set and edge set and , see Figure 2.
In this section, we obtaining new results for the OTIS networks .
Theorem 4. For network, we have
Proof. The has vertices and edges. Hence, there are two partitions, and . Hence, we have the following equalities:It can be easily seen that and .
By applying the definitions and equalities (16) and (17), we can write
Theorem 5. For network, we have
Proof. By applying the definitions and equalities (16) and (17), we have
Theorem 6. For network, we have
Proof. By applying the definitions and equalities (16) and (17), we can write
4. Graphical Representation and Discussion
In this paper, we discussed physical properties of some networks in terms of topological indices. The study of graphs and networks through topological descriptors area unit necessary to grasp their underlying topologies. Hence, in this paper, we computed general topological temperature indices. The graphical representations of general temperature indices of and area unit are represented in Figures 3 and 4.
5. Conclusion
In medical science, chemical, medical, biological, and pharmaceutical properties of molecular structure are essential for drug design. These properties can be studied by the topological index calculation. Hence, we have computed general topological indices of some networks such as networks and networks. We obtained the closed formulas of the general first and second temperature indices and the general temperature index for these networks. Our results can help to guess the many physical and chemical properties of networks.
Data Availability
The data involved in the examples of our manuscript are included within the article.
Disclosure
We would like to declare that the work described was original research that has not been published previously. This work was in memoriam of Dr. Rana Khoeilar, the author died prior to the submission of this paper. This is one of the last works of her.
Conflicts of Interest
The authors declare that they have no conflicts of interest.