Abstract
Polysaccharides are biomaterial with great biocompatibility, biodegradability, and low toxicity. There are long chains of monosaccharide units linked together by glycosidic bonds. They have a wide spectrum of functional properties and are essential to life’s survival. These are a makeup of storage polysaccharides (such as starch and glycogen). Starch is found in plants; the condensation of amylose and amylopectin produces starch. The main contribution of this paper is to compute Revan topological indices of Amylose and Amylopectin that can study their physicochemical characterization. Furthermore, an analysis was being carried out among topological indices to find out compatibility. These indices will be applicable in various useful research aspects.
1. Introduction
Chemical graph theory integrates graph theory and chemistry in an area of mathematics. To obtain insight into the physical features of chemical substances, graph theory is employed to mathematically model them. Chemical reaction network theory is a branch of applied mathematics aimed at simulating the performance of different chemical systems. It has gained an increasing research community since its inception in the 1960s, owing to its applications in biochemistry and theoretical chemistry.
Atoms are represented by vertices in chemical networks, and bonding is represented by vertices in chemical graphs theory. The topological index is the numerical parameter that predicts the characteristics of that chemical graph. A topological index (TI) is a mapping which gives a numerical value to a chemical graph based on several physicochemical properties of the chemical. Organic molecules’ biological activity, as well as their physicochemical and chemical properties, are linked to their molecular structure. Topological indices (TIs) and their derivatives, such as Quantitative Structure-Property Relationships (QSPRs), are widely used as molecular descriptors in the development of QSARs.
Quantitative Structure-Activity Relationships (QSARs), Quantitative Structure-Property Relationships (QSPRs), and Quantitative Structure-Toxicity Relationships (QSTRs) are all built using topological indices (TIs). The goal of establishing a topological index is to assign a numerical value to each chemical structure while keeping it as selective as possible. Structures can be classified and chemical and biological properties predicted using these indices [1, 2]. None of the known topological indices can uniquely characterize molecular networks due to the loss of information caused by the condensing of molecular topological properties into a single integer. Various indices, on the other hand, have been utilized to correlate various physicochemical and biological aspects. When it comes to mese indices, one important point to consider is how independent these indicators are from one another. Because different mathematical procedures [3, 4] are used to obtain distinct topological indices from the structure. Examining the extent of orthogonality among the various indexes is one technique to answer this question. In this article, be connected simple chemical structure, with vertices set and edges set. The degree of any vertex is denoted by . The edge between vertices and is denoted by . In present study, we calculate degree based Revan topological indices. Let is the minimum degree and is the maximum degree in . The revan degree of vertex is .
The topological index evolved from the Wiener index, in 1945, which Wiener defined in 1945 while researching alkane’s boiling point [5]. The first degree-based topological index represented by the Milan Randić index is the Randić index [6]. Revan and topological indices were introduced by Kulli in [7] for study see [8, 9].
The and hyper Revan indices were defined by Kulli [10] as follows:
The and modified Revan indices were introduced by Kulli in [11] as follows:
The sum connectivity Revan index was defined by Kulli in [12].
The product connectivity Revan index was defined by Kulli in [13].
The F-Revan was introduced as follows [14]:
The symmetric division Revan index is defined as follows. The study of [15] used for surface determination of polychlorobiphenyls [16] and formulated as follows:
The Hormonic Revan index is defined as follows [15]:
Inverse sum Revan index is [15]:
2. Formation and Result for Amylose Polysaccharide (Chain Structure)
Using the stretcher of Figure 1, we obtained Table 1.
Theorem 1. The first Revan index of Amylose polysaccharide (chain structure) is
Proof 1. Let
Theorem 2. The second Revan index of Amylose polysaccharide (chain structure) is
Proof 2. Let
Theorem 3. The hyper first Revan index of Amylose polysaccharide (chain structure) is
Proof 3. Let
Theorem 4. The second hyper Revan index of Amylose polysaccharide (chain structure) is
Proof 4. Let
Theorem 5. The modified first Revan index of Amylose polysaccharide (chain structure) is
Proof 5. Let
Theorem 6. The modified second Revan index of Amylose polysaccharide (chain structure) is
Proof 6. Let
Theorem 7. The harmonic Revan index of Amylose polysaccharide (chain structure) is
Proof 7. Let
Theorem 8. The sum division Revan index of Amylose polysaccharide (chain structure) is
Proof 8. Let
Theorem 9. The inverse Revan index of Amylose polysaccharide (chain structure) is
Proof 9. Let
3. Numerical and Graphical Representation
“The numerical and graphical representation of above-computed results are depicted in Figures 2 and 3.”
The graphical comparison of topological indices tells us how these are affected by the parameters involved. Furthermore, the behavior of each line tells us about the numerical value of the topological index that depends upon .
4. Formation and Result of Amylopectin Polysaccharide
Using the stretcher of Figure 4, we obtained Table 2.
Theorem 10. The first Revan index of Amylopectin Polysaccharide (branched chain) is
Proof 10. Let
Theorem 11. The second Revan index of Amylopectin Polysaccharide (branched chain) is
Proof 11. Let
Theorem 12. The hyper first Revan index of Amylopectin Polysaccharide (branched chain) is
Proof 12. Let
Theorem 13. The second hyper Revan index of Amylopectin Polysaccharide (branched chain) is
Proof 13. Let
Theorem 14. The modified first Revan index of Amylopectin Polysaccharide (branched chain) is
Proof 14. Let
Theorem 15. The modified second Revan index of Amylopectin Polysaccharide (branched chain) is
Proof 15. Let
Theorem 16. The harmonic Revan index of Amylopectin Polysaccharide (branched chain) is
Proof 15. Let
Theorem 17. The sum division Revan index of Amylopectin Polysaccharide (branched chain) is
Proof 17. Let
Theorem 18. The inverse Revan index of Amylopectin Polysaccharide (branched chain) is
Proof 18. Let
5. Numerical and Graphical Representation
“The numerical and graphical representation of above-computed results are depicted in the graphical representation are dedicated in Figures 5 and 6.”
The graphical comparison of topological indices tells us how these are affected by the parameters involved. Furthermore, the behavior of each line tells us about the numerical value of the topological index that depends upon .
6. Conclusion
In this manuscript, Revan vertex degree and degree-dependent Revan topological indices of Amylose and Amylopectin Polysaccharides are calculated. The Revan topological indices, for example, are topological descriptors that have been related with various chemical properties of molecules and have been shown to run parallel to preparing the boiling point and Kovats constants of the molecule. The research presented in this paper allows for a comparative evaluation of numerous topological indices proposed and employed in modern mathematical and theoretical chemistry. We focused our research on a small number of topological indices and only looked at Polysaccharides. Comparing various classes of descriptors (e.g., distance-based, entropies, and eigenvalue-based), comparing topological indices within different classes, and using sets of (molecular) graphs other than Polysaccharides to reach further conclusions are also tasks for the future. Moreover, these indices will be helpful in the analysis of the QSPR/QSAR and are used in the structure of the graph.
Data Availability
The data used to support the study are included in the paper.
Conflicts of Interest
The authors declare that there are no conflicts of interest.