# Physico-Chemical Characterization of Amylose and Amylopectin Using Revan Topological Indices

Authors: Muhammad Kamran; Sadik Delen; Riaz Hussain Khan; Nadeem Salamat; A. Q. Baig; Ismail Naci Cangul; Md. Ashraful Alam
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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, G be connected simple chemical structure, with VG vertices set and EG edges set. The degree of any vertex u is denoted by ℜ˜u. The edge between vertices u and v is denoted by uv. In present study, we calculate degree based Revan topological indices. Let δG is the minimum degree and ΔG is the maximum degree in G. The revan degree of vertex u is ru=ΔG+δG−ℜ˜G.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 1st and 2nd topological indices were introduced by Kulli in [7] for study see [8, 9].(1)R1G=∑uv∈EGru+rv,R2G=∑uv∈EGru⋅rv.The1st and 2nd hyper Revan indices were defined by Kulli [10] as follows:(2)HR1G=∑uv∈EGru+rv2,HR2G=∑uv∈EGru⋅rv2.The1st and 2nd modified Revan indices were introduced by Kulli in [11] as follows:(3)mR1G=∑uv∈EG1ru+rv,mR2G=∑uv∈EG1ru⋅rv.The sum connectivity Revan index was defined by Kulli in [12].(4)mR1G=∑uv∈EG1ru+rv.The product connectivity Revan index was defined by Kulli in [13].(5)mR1G=∑uv∈EG1ru⋅rv.TheF-Revan was introduced as follows [14]:(6)FRG=∑uv∈EGru2+rv2.The symmetric division Revan index is defined as follows. The study of [15] used for surface determination of polychlorobiphenyls [16] and formulated as follows:(7)SDDG=∑uv∈EGrurv+rvru.The Hormonic Revan index is defined as follows [15]:(8)HG=∑uv∈EG2ru+rv.Inverse sum Revan index is [15]:(9)IG=∑uv∈EGru⋅rvru+rv.


## 2. Formation and Result for Amylose Polysaccharide (Chain Structure)

Using the stretcher of Figure1, we obtained Table 1.Figure 1
Amylose polysaccharide (chain structure).Table 1
Degree-based edge partition using Figure1.
No. of edges1,21,32,33,3Frequencyn2n+25n−24nTheorem 1.
The first Revan index of Amylose polysaccharide (chain structure) is(10)R1G=36n+2.Proof 1.
Let(11)R1G=∑uv∈EGru+rvR1G=∑1,2∈EGru+rv+∑1,3∈EGru+rv+∑2,3∈EGru+rv+∑3,3∈EGru+rv=3+2n+3+12n+2+2+15n−2+1+14n=5n+42n+2+35n−2+24n=36n+2.Theorem 2.
The second Revan index of Amylose polysaccharide (chain structure) is(12)R2G=26n+2.Proof 2.
Let(13)R2G=∑uv∈EGru⋅rvR2G=∑1,2∈EGru⋅rv+∑1,3∈EGru⋅rv+∑2,3∈EGru⋅rv+∑3,3∈EGru⋅rv=3⋅2n+3⋅12n+2+2⋅15n−2+1⋅14n=6n+32n+2+25n−2+4n=26n+2.Theorem 3.
The hyper first Revan index of Amylose polysaccharide (chain structure) is(14)HR1G=118n+14.Proof 3.
Let(15)HR1G=∑uv∈EGru+rv2HR1G=∑1,2∈EGru+rv2+∑1,3∈EGru+rv2+∑2,3∈EGru+rv2+∑3,3∈EGru+rv2=3+22n+3+122n+2+2+125n−2+1+124n=25n+162n+2+95n−2+44n=118n+14.Theorem 4.
The second hyper Revan index of Amylose polysaccharide (chain structure) is(16)HR2G=78n+10.Proof 4.
Let(17)HR2G=∑uv∈EGru⋅rv2HR2G=∑1,2∈EGru⋅rv2+∑1,3∈EGru⋅rv2+∑2,3∈EGru⋅rv2+∑3,3∈EGru⋅rv2=3⋅22n+3⋅122n+2+2⋅125n−2+1⋅124n=36n+92n+2+45n−2+4n=78n+10.Theorem 5.
The modified first Revan index of Amylose polysaccharide (chain structure) is(18)Rm1G=13130n−16.Proof 5.
Let(19)Rm1G=∑uv∈EG1ru+rvRm1G=∑1,2∈EG1ru+rv+∑1,3∈EG1ru+rv+∑2,3∈EG1ru+rv+∑3,3∈EG1ru+rv=n3+2+2n+23+1+5n−22+1+4n1+1=n5+2n+24+5n−23+4n2=13130n−16.Theorem 6.
The modified second Revan index of Amylose polysaccharide (chain structure) is(20)Rm2G=223n−13.Proof 6.
Let(21)Rm2G=∑uv∈EG1ru⋅rvRm2G=∑1,2∈EG1ru⋅rv+∑1,3∈EG1ru⋅rv+∑2,3∈EG1ru⋅rv+∑3,3∈EG1ru⋅rv=n3⋅2+2n+23⋅1+5n−22⋅1+4n1⋅1=n6+2n+23+5n−22+4n=223n−13.Theorem 7.
The harmonic Revan index of Amylose polysaccharide (chain structure) is(22)HRG=13115n−13.Proof 7.
Let(23)HRG=∑uv∈EG2ru+rvHRG=∑1,2∈EG2ru+rv+∑1,3∈EG2ru+rv+∑2,3∈EG2ru+rv+∑3,3∈EG2ru+rv=2n3+2+22n+23+1+25n−22+1+24n1+1=2n5+2n+22+25n−23+4n=13115n−13.Theorem 8.
The sum division Revan index of Amylose polysaccharide (chain structure) is(24)SDRG=883n+53.Proof 8.
Let(25)SDRG=∑uv∈EGrurv+rvruSDRG=∑1,2∈EGrurv+rvru+∑1,3∈EGrurv+rvru+∑2,3∈EGrurv+rvru+∑3,3∈EGrurv+rvru=32+23n+31+132n+2+21+125n−2+11+114n=136n+1032n+2+525n−2+24n=883n+53.Theorem 9.
The inverse Revan index of Amylose polysaccharide (chain structure) is(26)IRG=24130n+16.Proof 9.
Let(27)IRG=∑uv∈EGru⋅rvru+rvIRG=∑1,2∈EGru⋅rvru+rv+∑1,3∈EGru⋅rvru+rv+∑2,3∈EGru⋅rvru+rv+∑3,3∈EGru⋅rvru+rv=3⋅23+2n+3⋅13+12n+2+2⋅12+15n−2+1⋅11+14n=65n+342n+2+235n−2+124n=24130n−16.


## 3. Numerical and Graphical Representation

“The numerical and graphical representation of above-computed results are depicted in Figures2 and 3.”Figure 2
Comparison ofR1G, R2G, HR1G and HR2G.Figure 3
Comparison ofRm1PF, Rm2PF, HR, IG and SDDG.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 uponn.


## 4. Formation and Result of Amylopectin Polysaccharide

Using the stretcher of Figure4, we obtained Table 2.Figure 4
Amylopectin Polysaccharide (branched chain).Table 2
Degree based edge partition forn>1 using Figure 4.
No. of edges1,32,22,33,3Frequency9n−4112n−1012n−8Theorem 10.
The first Revan index of Amylopectin Polysaccharide (branched chain) is(28)R1G=96n−58.Proof 10.
Let(29)R1G=∑uv∈EGru+rvR1G=∑1,3∈EGru+rv+∑2,2∈EGru+rv+∑2,3∈EGru+rv+∑3,3∈EGru+rv=3+19n−4+2+21+2+112n−10+1+112n−8=49n−4+41+312n−10+212n−8=96n−58.Theorem 11.
The second Revan index of Amylopectin Polysaccharide (branched chain) is(30)R2G=63n−36.Proof 11.
Let(31)R2G=∑uv∈EGru⋅rvR2G=∑1,3∈EGru⋅rv+∑2,2∈EGru⋅rv+∑2,3∈EGru⋅rv+∑3,3∈EGru⋅rv=3⋅19n−4+2⋅21+2⋅112n−10+1⋅112n−8=39n−4+41+212n−10+12n−8=63n−36.Theorem 12.
The hyper first Revan index of Amylopectin Polysaccharide (branched chain) is(32)HR1G=300n−170.Proof 12.
Let(33)HR1G=∑uv∈EGru+rv2HR1G=∑1,3∈EGru+rv2+∑2,2∈EGru+rv2+∑2,3∈EGru+rv2+∑3,3∈EGru+rv2=3+129n−4+2+221+2+1212n−10+1+1212n−8=169n−4+161+912n−10+412n−8=300n−170.Theorem 13.
The second hyper Revan index of Amylopectin Polysaccharide (branched chain) is(34)HR2G=141n−68.Proof 13.
Let(35)HR2G=∑uv∈EGru⋅rv2HR2G=∑1,3∈EGru⋅rv2+∑2,2∈EGru⋅rv2+∑2,3∈EGru⋅rv2+∑3,3∈EGru⋅rv2=3⋅129n−4+2⋅221+2⋅1212n−10+1⋅1212n−8=99n−4+161+412n−10+12n−8=141n−68.Theorem 14.
The modified first Revan index of Amylopectin Polysaccharide (branched chain) is(36)Rm1G=494n−9712.Proof 14.
Let(37)Rm1G=∑uv∈EG1ru+rvRm1G=∑1,3∈EG1ru+rv+∑2,2∈EG1ru+rv+∑2,3∈EG1ru+rv+∑3,3∈EG1ru+rv=9n−43+1+12+2+12n−102+1+12n−81+1=9n−44+14+12n−103+12n−82=494n−9712.Theorem 15.
The modified second Revan index of Amylopectin Polysaccharide (branched chain) is(38)Rm2G=21n−16912.Proof 15.
Let(39)Rm2G=∑uv∈EG1ru⋅rvRm2G=∑1,3∈EG1ru⋅rv+∑2,2∈EG1ru⋅rv+∑2,3∈EG1ru⋅rv+∑3,3∈EG1ru⋅rv=9n−43⋅1+12⋅2+12n−102⋅1+12n−81⋅1=9n−43+14+12n−102+12n−8=21n−16912.Theorem 16.
The harmonic Revan index of Amylopectin Polysaccharide (branched chain) is(40)HRG=492n−976.Proof 15.
Let(41)HRG=∑uv∈EG2ru+rvHRG=∑1,3∈EG2ru+rv+∑2,2∈EG2ru+rv+∑2,3∈EG2ru+rv+∑3,3∈EG2ru+rv=29n−43+1+212+2+212n−102+1+212n−81+1=29n−44+12+212n−103+12n−8=492n−976.Theorem 17.
The sum division Revan index of Amylopectin Polysaccharide (branched chain) is(42)SDRG=84n−1573.Proof 17.
Let(43)SDRG=∑uv∈EGrurv+rvruSDRG=∑1,3∈EGrurv+rvru+∑2,2∈EGrurv+rvru+∑2,3∈EGrurv+rvru+∑3,3∈EGrurv+rvru=31+139n−4+22+221+21+1212n−10+11+1112n−8=1039n−4+2+5212n−10+212n−8=84n−1573.Theorem 18.
The inverse Revan index of Amylopectin Polysaccharide (branched chain) is(44)IRG=834n−383.Proof 18.
Let(45)IRG=∑uv∈EGru⋅rvru+rvIRG=∑1,3∈EGru⋅rvru+rv+∑2,2∈EGru⋅rvru+rv+∑2,3∈EGru⋅rvru+rv+∑3,3∈EGru⋅rvru+rv=3⋅13+19n−4+2⋅22+21+2⋅12+112n−10+1⋅11+112n−8=349n−4+1+2312n−10+1212n−8=834n−383.


## 5. Numerical and Graphical Representation

“The numerical and graphical representation of above-computed results are depicted in the graphical representation are dedicated in Figures5 and 6.”Figure 5
Comparison ofR1G, R2G, HR1G and HR2G.Figure 6
Comparison ofRm1G, Rm2G, HR, IRG and SDDG.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 uponn.


## 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.



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