Document Type : Original Research Article
Authors
- Jyothi. MJ ^{} ^{} ^{1}
- K. Shivashankara ^{} ^{2}
^{1} Department of Mathematices, Maharanis Science College for Women, Mysore 570005, India
^{2} Department of Mathematics, Yuvarajas College, University of Mysore, Mysore 570005, India
Abstract
In this work, M_{hr}- polynomial of oxybenzone, menthyl anthranilate, benzophenone-4, and dimethylamino hydroxybenzoyl hexa benzoate were established. The degree-based topological indices HDR version of Modified Zagreb topological index (HDRM*), HDR version of Modified forgotten topological index (HDRF*), and HDR version of hyper Zagreb index (HDRHM*) were obtained. Accordingly, by using the derivative of M_{hr}- polynomial of sunscreens, the HDRM*, HDRF*, and HDRHM* topological indices of sunscreens were found.
Graphical Abstract
Keywords
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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