Comparative study of five topological invariants of supramolecular chain of different complexes of N-salicylidene-L-valine

1.   Introduction

The quantity of new drugs is growing daily due to the fast production of pharmaceuticals. Knowing the pharmacological, biological, and chemical features of these freshly manufactured medications is necessary to determine how effective they are against particular diseases. This is a laborious, time-consuming task that must be completed. When newly created medications fail to treat a particular condition, researchers work diligently to identify their shortcomings and increase their efficacy. This requires well-established labs, reagents, and equipment. However, there is a lack of funding in developing nations to set up appropriate labs with all necessary reagents and equipment to assess the effectiveness of these medications. We have learned from the literature that the molecular structures of drugs and their chemical and pharmacologic effects are inextricably linked. If we compute measurements of drug molecular structures and find topological indices, pharmaceutical researchers can determine their medicinal characteristics. Deficiencies in their chemical experiments and structures, as well as their medicine, can be eliminated as a result. Therefore, the mathematical methods used to compute topological indices are especially useful for developing nations since they can forecast the pharmacological, biological, and chemical properties of medications without requiring laborious labor or laboratory studies.

In mathematics, the chemical structure of a drug can be represented as a graph, where the edges and vertices of the bonded atoms' chemical bonds are depicted. Consider a drug's chemical structure to be represented by the graph $G$, which has certain edges (chemical bonds between atoms) and vertices (bonded atoms). The topological index, which assigns a specific value to each molecular structure, converts the molecular composition of a given medicine into a numerical value that is a real-valued function. In the past 40 years, scientists have created numerous significant indices to identify the characteristics of molecules. These topological indices have established their value by finding significant uses in molecular graphs, engineering, and nanomaterials.

We examined the v-phenylenic nanotubes and nanotori fourth particle bond availability file in ^[1]. The benzenoid circumcoronene series Hk's third-connectivity and third-sum-connectivity indices are examined in ^[2]. The characteristics of the Sombor index for some nanostructures are discussed in ^[3], the topological indices of NEPS of graphs were computed in ^[4], the Zagreb connection index for drugs related to specific chemical structures was investigated in ^[5], and face index for silicon carbides was covered in ^[6]. Reference ^[7] discussed Zagreb connection indices of silicate, hexagonal, oxide, and honeycomb networks. Predicted values of first Zagreb connection index for random cyclooctatetraene chain, random polyphenyl chain and random chain network are reported in ^[8]. In ^[9] the author investigated the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs. For more information, please refer to ^{[10,11,12,13,14,15,16,17,18]}. Twenty different amino acids are used to make proteins, and one of them is Lv, an essential amino acid for mammals. From Lv and salicylaldehyde, a chiral Schiff base carboxylate ligand known as NsLv is created, which has a remarkable capacity for coordination and many coordination modes toward metal ions. This ligand produces numerous metal complexes that are extensively used in research, particularly with regard to their characteristics and molecular structures ^{[19,20,21,22,23]}. These metal complexes are widely used as enantio-selective reaction catalysts, efficient reagents for DNA cleavage, and chiral fluorescent molecular sensors ^[24]. Organotin compounds, which are based on tin bonded to carbon, are organometallic compounds with a wide range of uses in materials science, organic synthesis, medicine and catalysis. Moreover, organotin compounds with carboxylate ligands have become compounds of great interest due to effective catalytic and cytotoxic activity along their various structures ^[25,26,27]. Organotin NsLv complexes exhibit significant biological and optical properties. They can be synthesized by combining sodium and potassium NsLv with either an oxide or a chloride ^[22,28,29]. Dialkyltin complexes of N-salicylidene-L-valine can be synthesized by heating and refluxing L-valine, salicylaldehyde, dialkyltin dichloride and DBU (1, 8-diazabicyclo [5.4.0] undec-7-ene) in methanol for 4 hours in a round bottom flask. In the process, the remaining yellow residue is rinsed with water to dilute the Schiff base ligand, DBU salt N-salicylidene-L-valinate, before it is evaporated by rotary evaporator in order to produce the pure product. For deeper information about the mechanism and synthesis of this structure, see ^[34].

Definitions

The reduced second Zagreb index ^[30] is defined as

The reciprocal Randi$\acute{c}$ index ^[31] is defined as

The first and second Zagreb indices ^[32] are defined as

The augmented Zagreb index ^[33] is defined as

2.   Sc in dialkyltin complex-2, 3, 4 of NsLv

The four types of complexes found in the Sc of dialkyltin are NsLv complexes 2, 3, 4 and 5. Complexes $C_{2, \varPsi}$, $C_{3, \varPsi}$ and $C_{4, \varPsi}$ will be covered in this section. The extended molecular structure unit graph shown in Figure 1 was created from the compound construction shown in Figure 2 ^[34]. Figure 7 shows the molecular structure's extended unit graph derived from the chemical structure shown in Figure 8, while Figure 4 displays the extended molecular structure unit graph created based on the chemical structure shown in Figure 5. The $C_{2, \varPsi}$ have 1, 2, 3, 4 and 5 degree bonds. The $C_{3, \varPsi}$ have 1, 2, 3 and 5 degree bonds. The $C_{4, \varPsi}$ have 1, 2, 3 and 5 degree bonds. In addition, Tables 1, 3 and 5 display the edge type, degree of the end vertices-and frequency of the edge, all of which are extremely helpful to our computational work. The arrangement and size of $C_{2, \varPsi}$- are $|V(C_{2, \varPsi})| = 21\varPsi$, $|E(C_{2, \varPsi})| = 25\varPsi-2$, respectively. The order and size of $C_{3, \varPsi}$- are $|V(C_{3, \varPsi})| = 26\varPsi$, $|E(C_{3, \varPsi})| = 30\varPsi-1$, respectively. The order and size of $C_{4, \varPsi}$- are $|V(C_{4, \varPsi})| = 28\varPsi$, $|E(C_{4, \varPsi})| = 31\varPsi-1$, respectively.

Theorem 1. Let $C_{2, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-2, then the reduced second Zagreb index is,

Proof. By referring to the definition of reduced second Zagreb index and Table 1,

After simplifying the preceding statement, we get the following result:

□

Theorem 2. Let $C_{2, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-2, then the reciprocal Randi$\acute{c}$ index is,

Proof. By referring to the definition of reciprocal Randi$\acute{c}$ index and Table 1,

After simplifying the preceding statement, we get the following result:

□

Theorem 3. Let $C_{2, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-2, then the first Zagreb index is,

Proof. By referring to the definition of first Zagreb index and Table 1,

After simplifying the preceding statement, we get the following result:

□

Theorem 4. Let $C_{2, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-2, then the second Zagreb index is,

Proof. By referring to the definition of second Zagreb index and Table 1,

After simplifying the preceding statement, we get the following result:

□

Theorem 5. Let $C_{2, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-2, then the augmented Zagreb index is,

Proof. By referring to the definition of augmented Zagreb index and Table 1,

After simplifying the preceding statement, we get the following result:

□

Theorem 6. Let $C_{3, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-3, then the reduced second Zagreb index is,

Proof. By using the definition of reduced second Zagreb index and Table 3,

After simplifying the preceding statement, we get the following result:

□

Theorem 7. Let $C_{3, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-3, then the reciprocal Randi$\acute{c}$ index is

Proof. By referring to the definition of reciprocal Randi$\acute{c}$ index and Table 3,

After simplifying the preceding statement, we get the following result:

□

Theorem 8. Let $C_{3, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-3, then the first Zagreb index is

Proof. By referring to the definition of first Zagreb index and Table 3,

After simplifying the preceding statement, we get the following result:

□

Theorem 9. Let $C_{3, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-3, then the second Zagreb index is

Proof. By referring to the definition of second Zagreb index and Table 3,

After simplifying the preceding statement, we get the following result:

□

Theorem 10. Let $C_{3, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-3, then the augmented Zagreb index is

Proof. By referring to the definition of augmented Zagreb index and Table 3,

After simplifying the preceding statement, we get the following result:

□

Theorem 11. Let $C_{4, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-4, then the reduced second Zagreb index is

Proof. By referring to the definition of reduced second Zagreb index and Table 5,

After simplifying the preceding statement, we get the following result:

□

Theorem 12. Let $C_{4, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-4, then the reciprocal Randi$\acute{c}$ index is

Proof. By referring to the definition of reciprocal Randi$\acute{c}$ index and Table 5,

After simplifying the preceding statement, we get the following result:

□

Theorem 13. Let $C_{4, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-4, then the first Zagreb index is

Proof. By referring to the definition of first Zagreb index and Table 5,

After simplifying the preceding statement, we get the following result:

□

Theorem 14. Let $C_{4, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-4, then the second Zagreb index is

Proof. By referring to the definition of second Zagreb index and Table 5,

After simplifying the preceding statement, we get the following result:

□

Theorem 15. Let $C_{4, \varPsi}$, with $\varPsi \geq1$ is a Sc graph in NsLv dialkyltin complex-4, then the augmented Zagreb index is

Proof. By referring to the definition of augmented Zagreb index and Table 5,

After simplifying the preceding statement, we get the following result:

□

3.   Discussion

For the purpose of examining topological indices, chemical structures and their designed networks of vertices and edges are thought to be useful tools. You can observe how a particular structure and graph behave using topological indices, which offer numerical values. The five degree-based topological indices of our table-based structure that are the subject of the report are examined. Corresponding graphs were created using parameter values ranging from 1 to 7 for each of the five topological indices. Figure 3 plots five degree-based topological indices for the structure of the Sc in dialkyltin complex-2 of NsLv ($C_{2, \varPsi}$) based on the information in Table 2. The plots grow in size linearly. The reciprocal Randi$\acute{c}$ index and the reduced second Zagreb index are close to each other, but overall the reciprocal Randi$\acute{c}$ index moves sharply in size among the others in ($C_{2, \varPsi}$). The structure of the Sc in dialkyltin complex-3 of NsLv ($C_{3, \varPsi}$) is depicted in Figure 6 as a plot of five degree based topological indices derived from Table 4 data. The plots expand in size in a linear fashion. The reciprocal Randi$\acute{c}$ index and the second-reduced Zagreb index are close to one another, but generally the reciprocal Randi$\acute{c}$ index moves sharply in size among the others in ($C_{3, \varPsi}$). Because of the structure of the Sc in dialkyltin complex-4 of NsLv ($C_{4, \varPsi}$) data from Table 6 are used to generate a plot of five degree-based topological indices in Figure 9. The plots enlarge linearly. The reciprocal Randi$\acute{c}$ index and the reduced second Zagreb index are close together, however, in comparison to the others, the scaled-down second Zagreb index moves dramatically in size. ($C_{4, \varPsi}$). It makes sense to assume that when examining our chosen structures ($C_{2, \varPsi}$) and ($C_{3, \varPsi}$), the reciprocal Randi$\acute{c}$ index is preferable to the others, but when examining ($C_{4, \varPsi}$) the reduced second Zagreb index is more appropriate for study than the others.

4.   Conclusions

The chemical structure known as a Sc in the dialkyltin complexes-2, 3 and 4 of NsLv was examined in this study using computational methods. We investigated some relevant topological descriptors, such as the reduced second Zagreb index, the reciprocal Randi$\acute{c}$ index, the first and second Zagreb index- and the augmented Zagreb index, to examine the behavior of our selected structures. As future work, many other degree and distance based entropies such as Sombor based, eccentricity based and Wiener based entropies can be computed for these structures.

Conflict of interest

The authors declare there is no conflict of interest.