# On the Temperature Indices of Molecular Structures of Some Networks

Authors: Akbar Jahanbani; Rana Khoeilar; Murat Cancan
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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 graphG, we denoted the vertex set by VG=x1,x2,x3,…,xn and the edge set by ξG,ξG=ε. The degree of the vertex x∈VG is denoted by ζx.For any graphsG with n vertices, the temperature of a vertex x is defined in [13] as(1)ψx=ζxn−ζx.Kulli in [14] defined the following topological indices asThe general first temperature index(2)ψ1ωG=∑xy∈ξG(ψx+ψy)ω.The general second temperature index(3)ψ2ωG=∑xy∈ξG(ψx×ψy)ω.The general temperature index(4)ψωG=∑xy∈ξGψxω+ψyω,where ω∈R.


## 2. Results forNPn Network

In this section, we compute the exact formulas for the OTIS networksNPn of some general temperature indices, and we also examine the relationships between these topologies with graphical diagrams.LetPn be the path of n vertices and NPn be OTIS (swapped) network with basis network Pn, see Figure 1.Figure 1
The graph ofNP7.We start by computing the general first temperature index.Theorem 1.
ForNPn network, we have(5)ψ1ωNPn=21n2−1+3n2−3ω+34n2−2ω+6n−142n2−2+3n2−3ω+3n−2n−326n2−3ω.Proof.
The OTIS networksNPn have n2 vertices and 3n2−n/2, edges. Hence, there are three partitions, V1=y∈VNPn|ζy=1, V2=y∈VNPn|ζy=2 and V3=y∈VNPn|ζy=3. Hence, we can write that(6)ξ1=ε=xy∈ξNPn|ζx=1andζy=3,(7)ξ2=ε=xy∈ξNPn|ζx=2andζy=2,(8)ξ3=ε=xy∈ξNPn|ζx=2andζy=3,(9)ξ4=ε=xy∈ξNPn|ζx=3andζy=3.
It can be easily seen thatξ1=2,ξ2=3, ξ3=6n−14 and ξ4=3n−2n−3/2.
By applying the definitions and Equalities (6)–(9), we can write(10)ψ1ωNPn=∑xy∈ξNPnψx+ψyω=∑xy∈ξ1NPnψx+ψyω+∑xy∈ξ2NPnψx+ψyω+∑xy∈ξ3NPnψx+ψyω+∑xy∈ξ4NPnψx+ψyω=21n2−1+3n2−3ω+32n2−2+2n2−2ω+6n−142n2−2+3n2−3ω+3n−2n−323n2−3+3n2−3ω=21n2−1+3n2−3ω+34n2−2ω+6n−14+2n2−2+3n2−3ω+3n−2n−326n2−3ω.Now, we compute the general second temperature index.Theorem 2.
ForNPn network, we have(11)ψ2ωNPn=23n2−1n2−3ω+34n2−22ω+6n−146n2−2n2−3ω+3n−2n−329n2−32ω.Proof.
By applying the definitions and equalities (6)–(9), we have(12)ψ2ωNPn=∑xy∈ξNPn(ψx×ψy)ω=∑xy∈ξ1NPn(ψx×ψy)ω+∑xy∈ξ2NPn(ψx×ψy)ω+∑xy∈ξ3NPn(ψx×ψy)ω+∑xy∈ξ4NPn(ψx×ψy)ω=21n2−1×3n2−3ω+32n2−2×2n2−2ω+6n−142n2−2×3n2−3ω+3n−2n−323n2−3×3n2−3ω=23n2−1n2−3ω+34n2−22ω+6n−146n2−2n2−3ω+3n−2n−329n2−32ω.Here, we compute the general temperature index.Theorem 3.
ForNPn network, we have(13)ψωNPn=3n23n2−3ω+6n2n2−2ω−9n3n2−3ω+21n2−1ω+63n2−3ω−82n2−2ω.Proof.
By applying the definitions and Equalities (6)–(9), we can write(14)ψωNPn=∑xy∈ξNPnψxω+ψyω=∑xy∈ξ1NPnψxω+ψyω+∑xy∈ξ2NPnψxω+ψyω+∑xy∈ξ3NPnψxω+ψyω+∑xy∈ξ4NPnψxω+ψyω=21n2−1ω+3n2−3ω+32n2−2ω+2n2−2ω+6n−142n2−2ω+3n2−3ω+3n−2n−323n2−3ω+3n2−3ω=3n23n2−3ω+6n2n2−2ω−9n3n2−3ω+21n2−1ω+63n2−3ω−82n2−2ω.


## 3. Results forNTt Network

The OTIS (swapped) networkNTt is derived from the graph Υ, which a graph with vertex set VNTΥ=z,x|z,x∈VΥ and edge set ξ(NTΥ)=z,x1,z,x2z∈VΥ,(x1,x2)∈ξ(NTΥ)∪(z,x,z,x,z)|z,x∈V(Υ)) and z≠x, see Figure 2.Figure 2
The graphNT7.In this section, we obtaining new results for the OTIS networksNTt.Theorem 4.
ForNTt network, we have(15)ψ1ωNTt=nttn+22−t+t+1n+22−t−1ω+2ω−1t+1ωn2t+1−n2t+1n+22−t−1ω.Proof.
TheNTt has n+22 vertices and n3/2+3n2+11n/2+3 edges. Hence, there are two partitions, V1=y∈VNTt|ζy=t and V2=y∈VNPn|ζy=t+1. Hence, we have the following equalities:(16)ξ1=ε=xy∈ξNTt|ζx=tandζy=t+1,(17)ξ2=ε=xy∈ξNTt|ζx=t+1andζy=t+1.
It can be easily seen thatξ1=nt and ξ2=n2t+1−n2t+1/2.
By applying the definitions and equalities (16) and (17), we can write(18)ψ1ωNTt=∑xy∈ξNTt(ψx+ψy)ω=∑xy∈ξ1NTt(ψx+ψy)ω+∑xy∈ξ2NTt(ψx+ψy)ω=nttn+22−t+t+1n+22−t−1ω+n2t+1−n2t+12t+1n+22−t−1+t+1n+22−t−1ω=nttn+22−t+t+1n+22−t−1ω+n2t+1−n2t+122t+1n+22−t−1ω=nttn+22−t+t+1n+22−t−1ω+2ω−1t+1ωn2t+1−n2t+1n+22−t−1ω.Theorem 5.
ForNTt network, we have(19)ψ2ωNTt=nttt+1n+22−tn+22−t−1ω+n2t+1−n2t+12t+1n+22−t−12ω.Proof.
By applying the definitions and equalities (16) and (17), we have(20)ψ2ωNTt=∑xy∈ξNTt(ψx×ψy)ω=∑xy∈ξ1NTt(ψx×ψy)ω+∑xy∈ξ2NTt(ψx×ψy)ω=nttn+22−t×t+1n+22−t−1ω+n2t+1−n2t+12t+1n+22−t−1×t+1n+22−t−1ω=nttn+22−t×t+1n+22−t−1ω+n2t+1−n2t+12t+12n+22−t−12ω=nttt+1n+22−tn+22−t−1ω+n2t+1−n2t+12t+1n+22−t−12ω.Theorem 6.
ForNTt network, we have(21)ψωNTt=nttn+22−tω+t+1n+22−t−1ω+n2t+1−n2t+1t+1n+22−t−1ω.Proof.
By applying the definitions and equalities (16) and (17), we can write(22)ψωNTt=∑xy∈ξNTtψxω+ψyω=∑xy∈ξ1NTtψxω+ψyω+∑xy∈ξ2NTtψxω+ψyω=nttn+22−tω+t+1n+22−t−1ω+n2t+1−n2t+12t+1n+22−t−1ω+t+1n+22−t−1ω=nttn+22−tω+t+1n+22−t−1ω+n2t+1−n2t+1t+1n+22−t−1ω.


## 4. Graphical Representation and Discussion

In this paper, we discussed physical properties of someOTIS 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 NPn and NTt area unit are represented in Figures 3 and 4.Figure 3
Comparison ofψ1ω,ψ2ω, and ψω for NPn.Figure 4
Comparison ofψ1ω,ψ2ω and ψω for NTt.


## 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 someOTIS networks such as NPn networks and NTt 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.



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