Introduction

A molecular graph is an undirected graph. It is denoted by G=(V, E), which shows the general properties of the molecular compound, where the vertices of the shovel |V| represent the number of atoms. In contrast, the edges E represent the juxtaposition relationship between those vertices that represent the atoms of the molecular compound [1]. In a molecular graph, the vertices represent atoms, and the edges represent chemical bonds, and they correspond to them. Suppose the G graph represents a chemical compound that contains a set of vertices V(G), and a set of edges is E(G). In that case, the degree of the vertex can be defined as the number of neighbors of the vertex in and is represented define by d_G(v), which is the total number of edges associated with v [2]. Topological indices are essential relationships in modeling quantitative relationships for the structure and activity of chemical graphs [3]; for more didcestion about topological indices [7-30].

Topological indicators are fixed values in a graph by which the real values are assigned, taking the graph as a consistent median and giving similar graphs the same value. Wiener index gave the first topological index [2].

In 1947, for studying boiling points of alkanes, one of the topological indices invented at the preliminary level is the so-called Zagreb index, first provided by [4,5]. They investigated how an electron whole energy relied on the shape of molecules and became discussed. The primary Zagreb indices 

DAlsinai A, at al, [1] define (1):

HDR of Zagreb index version modified of Hyper HDR Zagreb indices modified forgotten topological index of graph G by:

Materials and methods

Our main results including comuting M_hr- polynomial of Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate, the degree-based topological indices HDRM*, HDRF*, and HDRHM* obtained by using the derivative of M_hr- polynomial of sunscreens the HDRM*,  HDRF*, and HDRHM* topological indices of sunscreens, are found.

Main results

This section includes two subsections computation Mhr- polynomial of Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate, and computation topological indices of sunscreens (Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate).

Evalute the - polynomial of Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate

In this subsection, we compute the Polynomial of the sunscreens structures and we represent Mhr (3D) graphical using MATLAB.

Oxybenzone

Let  be a graph of oxybenzone, as shown in Figure 1. Then the graph 𝐺 has 17 vertices and 17 edges.

Theorem 1. Let 𝐺 be a graph of oxybenzone. Then

Proof. Let G a graph of Oxybenzone as in Figure 1. Then by using Table 2 and Definition2, we have:

Menthyl anthranilate

Let G be a graph of Menthyl Anthranilate, as shown in Figure 3. The graph 𝐺 contain 18 vertices and 19 edges.

Theorem 2. Let 𝐺 be a graph of menthyl anthranilate. Then

Proof. Let G be a graph of Menthyl Anthranilate as in Figure3. Then by using Table 3 and Definition 2, we have

Benzophenone-4

Let 𝐺 be a graph of Benzophenone-4, as shown in Figure 5. Then the graph 𝐺 contain 21 points and 22 edges and

Theorem 3. Let 𝐺 be the graph of benzophenone-4. Then

Proof. Let G be the graph of Benzophenone-4, as seen in Figure 5. Then by using Table 4 and Definition 2, we have:

Dimethylamino hydroxybenzoyl hexa benzoate

Let 𝐺 be the graph of Dimethylamino hydroxybenzoyl hexa benzoate, as shown in Figure 7. The graph 𝐺 contains 21 points and 22 edges.

Theorem 4. Let 𝐺be a moleculargraph of Dimethylamino hydroxybenzoyl hexa benzoate, then

Proof. Let Ga molecular graph of Dimethylamino hydroxybenzoyl hexa benzoate, as seen in Figure 7. Then by using Table 5 and Definition 2, we have

Compution HDR topological indices of sunscreens (Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate)

In this subsection, we computed HDR indices of Modified Zagreb topological index (HDRM*), (HDRF*), and (HDRHM*)   by using the derivative of Mhr- polynomial of sunscreens

Theorem 5. Let 𝐺 be the graph of Oxybenzone, and 

Using Table 1, we find the outcome.■

Theorem 6. Let 𝐺 be a molecular graph of Menthyl Anthranilate, and 

Using Table 1, we find the outcome .■

Theorem 7. Let 𝐺 be a molecular graph of Benzophenone-4, and 

Using Table 1, we find the outcome   .■

Theorem 8. Let 𝐺 be a molecular graph of Dimethylamino hydroxybenzoyl hexa benzoate, and 

Conclusion

We found Mhr- polynomial of Oxybenzone, Menthyl Anthranilate, Benzophenone-4, and Dimethylamino hydroxybenzoyl hexa benzoate, the degree-based topological indices  HDRM*, HDRF*, HDRHM*  obtained. By using the derivative of Mhr- polynomial of sunscreens the HDRM*, HDRF*, and HDRHM* topological indices of sunscreens, were found. The results of the HDRM*, HDRF*, and HDRHM* of sunscreens are better than the leap indices and Zagreb Indices.

Orcid:

Jyoth. MJ: https://orcid.org/0000-0002-4682-4713

Shivashankara: https://orcid.org/0000-0001-9473-0474

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How to cite this article: Jyothi. MJ*, K. Shivashankara. Topological properties of sunscreens using mhr-polynomial of graph. Eurasian Chemical Communications, 2022, 4(5), 402-410. Link: http://www.echemcom.com/article_145641.html

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