# Degree-Based Topological Indices of Polysaccharides: Amylose and Blue Starch-Iodine Complex

Authors: Anam Rani; Usman Ali
Journal: Journal of Chemistry
 (2021)

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## Abstract

Starch is a polymer of glucose where alpha-linkages are associated with glucopyranose units. It comprises a mixture of amylose and amylopectin. Furthermore, amylose is a linear chain of hundreds of glucose molecules. Starches are not allowed to be dissolved in water. They can be digested by breaking down alpha bonds (glycosidic bonds). Its cyclic degradation products, called cyclodextrins, are the best role models for amylose. They can be considered simple turns of the amylose propeller that has imploded into a circular path. Both humans and animals have amylases, which allow them to digest starches. The important sources of starch include potatoes, rice, wheat, and maize for human consumption. The production of starches is how plants store glucose. The blue colour of starch produced by an iodine solution or iodine reaction is used for its identification. Polysaccharides with a reduced degree of polymerization, known as dextrins, are produced in the starch’s partial acid hydrolysis. Complete hydrolysis leads to glucose. In this article, we compute the topological properties: Zagreb indexM1Γ and M2Γ, Randić index RαΓ for α=−1/2,−1,1/2,1, atom-bond connectivity index ABCΓ, geometric-arithmetic index GAΓ, fourth atom-bond connectivity index ABC4Γ, fifth geometric-arithmetic index GA5Γ, and degree-based topological indices of a graph Γ representing polysaccharides, namely, amylose and blue starch-iodine complex. In the end, we compare these indices and depict their graphic behavior.


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## Body

## 1. Introduction

Amylose has the most basic structure of all nutritional polysaccharides, composed purely of glucose polymers connected only byα1−4 bonds. Notice that starch is, in fact, a combination of amylose and amylopectin. Amylose is not allowed to be dissolved in water and is more difficult to digest compared to amylopectin. The complexing of amylopectin with amylose facilitates its water—another view of amylose solubility and digestibility. Amylose plays an important role in the storage of plant energy, and as plants do not require glucose to explode, its dense structure and slow breakdown features are under plant’s growth. Another function of polysaccharides within cells refers to structural support. Besides, hemicelluloses are another group of polysaccharides located in plant cell walls.In 1814, Colin and Claubry discovered the starch-iodine reaction, which is well renowned to any chemist from basic courses in qualitative and quantitative analysis.The first topological index was derived in 1947 when Wiener worked on the boiling point of paraffin alkanes. It was known as the Wiener number. Later on, it is called a path number. The work [1] described the M-polynomial and degree-based topological indices of graphs. The authors in [2, 3] discussed the symmetric divisor deg index of graphs, first Zagreb after 30 years in changed form and topological indices of molecular structure. The authors in [4] also discussed the π electron energy of hydrocarbons. In recent years, Hasni et al. computed the degree-based topological indices of the line graph of benzene ring embedded in P-type surface in the 2D network [5]. In [6], the authors calculated the index numbers for the edge version of the geometric-arithmetic index of nanocones. Much research has been done to explain the nature of chromophore absorption at 620 nm that yields starch-iodine complex, the distinctive dark blue colour. Still, there seem to have been many disputes that might be addressed to some extent in recent decades.LetΓ be connected simple graph with VΓ a set of vertices and EΓ a set of edges. Let u∈VΓ and its degree is represented by ℜu˜. The idea of degree-based topological indices began from Wiener index; in 1945, Wiener defined them while studying alkane’s boiling point cf. [7]. The first degree-based topological index is Randić index given by Milan Randić in [8] and is described as(1)R−1/2=∑uv∈EΓ1ℜu˜ℜv˜.Generalized Randić index (denoted asRαΓ) is described as follows:(2)RαΓ=∑uv∈EΓℜu˜ℜv˜α,α=1,12,−12,−1.Inverse generalized Randić index (denoted asRRαΓ) is described as(3)RRαΓ=∑uv∈EΓ1ℜu˜ℜv˜α.In [4, 9, 10], Gutman and Trinajstić introduced and defined the first Zagreb index (denoted as M1Γ) and second Zagreb index (denoted as M2Γ) as(4)M1Γ=∑uv∈EΓℜu˜+ℜv˜,M2Γ=∑uv∈EΓℜu˜ℜv˜.In [11], Estrada introduced and studied about the atom-bond connectivity index (denoted as ABCΓ). It is defined as follows:(5)ABCΓ=∑uv∈EΓℜu˜+ℜv˜−2ℜu˜ℜv˜.Geometric-arithmetic index (denoted asGAΓ) was given by Vukicˇevic´ cf. [12] and is defined as follows:(6)GAΓ=∑uv∈EΓ2ℜu˜ℜv˜ℜu˜+ℜv˜.The fourth version of the ABC index (denoted asABC4Γ) was introduced by Ghorbani in [13] and is defined as(7)ABC4Γ=∑uv∈EΓSu+Sv−2SuSv,where Su=∑v=NΓuℜv˜ and NΓu=v∈VΓ|uv∈EΓ.The fifth version of the GA index (denoted asGA5Γ) was given by Graovac cf. [14] and is defined as(8)GA5Γ=∑uv∈EΓ2SuSvSu+Sv.


## 2. Result for Amylose

Starch is a polymer of glucose whose glucopyranose alpha bonds bind cells. It is a mixture of amylose and amylopectin. Amylose is a linear chain of hundreds of glucose molecules. Starches cannot be dissolved in water. They can be digested by breaking the alpha bonds (glycosidic bonds). Amylose is a polysaccharide composed ofα-D-glucose units, linked by α1−4 glycosidic bonds. It is one of the two starch components that make up about 20 to 30 percent. Due to its tight spiral structure, amylose seems to be more resilient to digestion than other starch molecules and is, thus, a significant form of resistant starch [15] (see Figure 1 for a molecular structure of amylose and Figure 2 for its unit graph and the graph model corresponding to amylose for n = 4, where n is the number of units). In amylose, there are three types of vertices having degrees 1, 2, and 3. For n≥2, amylose has four types of edge partitions as(9)E1,2Γ=ℜu˜=1,ℜv˜=2andu,v∈VΓ,E1,3Γ=ℜu˜=1,ℜv˜=3andu,v∈VΓ,E2,3Γ=ℜu˜=2,ℜv˜=3andu,v∈VΓ,E3,3Γ=ℜu˜=3,ℜv˜=3andu,v∈VΓ.Figure 1
Molecular structure of amylose.Figure 2
(a) Graph of amylose forn = 1 and (b) graph of amylose for n = 4.Theorem 1.
For alln≥2, let Γ be the graph of amylose, then we have the following:(10)R1Γ=74n−6,R1/2Γ=29.1258n−1.4349,R−1/2Γ=5.2363n+0.3382,R−1Γ=2.4444n+0.3334.Proof.
The general Randić connectivity indexRαΓ for α=1 is(11)R1Γ=∑uv∈EΓℜu˜ℜv˜.
From Table1 and equation (2), we get(12)R1Γ=n1×2+2n+21×3+5n−22×3+4n3×3=74n−6.
Now, forα=1/2, the general Randić connectivity index RαΓ is(13)R1/2Γ=∑uv∈EΓℜu˜ℜv˜.
Again, from Table1 and equation (2), we have(14)R1/2Γ=n1×2+2n+21×3+5n−22×3+4n3×3=29.1258n−1.4349.
Ifα=−1/2, then(15)R−1/2Γ=∑uv∈EΓ1ℜu˜ℜv˜.
From Table1 and equation (2), it follows that(16)R−1/2Γ=n1×2+2n+21×3+5n−22×3+4n3×3=5.2363n+0.3382.
Now, forα=−1, we have(17)R−1Γ=∑uv∈EΓ1ℜu˜ℜv˜.
From Table1 and equation (2), we get(18)R−1Γ=n1×2+2n+21×3+5n−22×3+4n3×3=2.4444n+0.3334.Table 1
Edge partition of edges based on the degree of vertices.
Types of edgesE1,2E1,3E2,3E3,3Edges(1, 2)(1, 3)(2, 3)(3, 3)Frequencyn2n+25n−24nTheorem 2.
For alln≥2, let Γ be the graph of amylose, then, we have the following:(19)M1Γ=60n−2,M2Γ=74n−6,ABCΓ=8.5423n+0.2188,GAΓ=11.5738n−0.2276.Proof.
By using Table1 and equation (4), we get(20)M1Γ=∑uv∈EΓℜu˜+ℜv˜=n1+2+2n+21+3+5n−22+3+4n3+3=60n−2,M2Γ=∑uv∈EΓℜu˜ℜv˜=n1×2+2n+21×3+5n−22×3+4n3×3=74n−6.M2Γ=∑uv∈EΓℜu˜ℜv˜=n1×2+2n+21×3+5n−22×3+4n3×3=74n−6.
By using Table1 and equation (5), we get(21)ABCΓ=∑uv∈EΓℜu˜+ℜv˜−2ℜu˜ℜv˜=n1+2−21×2+2n+21+3−21×3+5n−22+3−22×3+4n3+3−23×3=8.5423n+0.2188.
By using Table1 and equation (6), we get(22)GAΓ=∑uv∈EΓ2ℜu˜ℜv˜ℜu˜+ℜv˜=2n1×21+2+2n+21×31+3+5n−22×32+3+4n3×33+3=11.5738n−0.2276.
In Table2, we give the edge partition centered on degree sum of end vertices for each edge.Table 2
Edge partition based on the degree sum of end vertices of each edge.
Types of edgesE2,4E3,6E3,7E4,7E6,6E6,7E6,8E7,7E7,8Number of edges(2, 4)(3, 6)(3, 7)(4, 7)(6, 6)(6, 7)(6, 8)(7, 7)(7, 8)Frequencyn12n+1n13n−1n−12n+12n−2Theorem 3.
For alln≥2, let Γ be the graph of amylose, then we have(23)ABC4Γ=6.4972n+0.2874,GA5Γ=11.7142n−0.123.Proof.
By using Table2 and equation (7), we get(24)ABC4Γ=∑uv∈EΓSu+Sv−2SuSv=n2+4−22×4+3+6−23×6+2n+13+7−23×7+n4+7−24×7+6+6−26×6+3n−16+7−26×7+n−16+8−26×8+2n+17+7−27×7+2n−27+8−27×8=6.4972n+0.2874.
By using Table2 and equation (8), we get(25)GA5Γ=∑uv∈EΓ2SuSvSu+Sv=2n2×42+4+3×63+6+2n+13×73+7+n4×74+7+6×66+6+3n−16×76+7+n−16×86+8+2n+17×77+7+2n−27×87+8=11.7142n−0.123.


## 3. Numerical and Graphical Representation

The numeric representation of the results calculated above is illustrated in Tables3 and 4 , while the graphic representation is devoted to Figures 3 and 4 .Table 3
Numerical comparison ofM1Γ, M2Γ, ABCΓ, GAΓ, R1Γ, R−1Γ, R1/2Γ, and R−1/2Γ.
nM1ΓM2ΓABCΓGAΓR1ΓR−1ΓR1/2ΓR−1/2Γ158688.761111.3462682.777827.69095.5746211814217.303422.921425.222256.816610.8109317821625.845734.49382167.666785.942416.0474423829034.38846.067629010.1111115.068221.2837529836442.930357.641436412.5556144.193926.5201635843851.472669.215243815173.319731.7565741851260.014980.78951217.4444202.445536.9929847858668.557292.362858619.8889231.571242.2293953866077.0995103.936666022.3333260.696947.46561059873485.6418115.510473424.7778289.822852.7020Table 4
Numerical comparison ofABC4Γ and GA5Γ.
n12345678910ABC4Γ6.78513.28219.77926.27632.77339.27145.76852.26558.76265.259GA5Γ11.59123.30535.01946.73458.44870.16281.87693.5906105.305117.019Figure 3
(a) Comparison ofRα for α=1, −1, 1/2, −1/2 and (b) comparison of M1Γ, M2Γ, ABCΓ, and GAΓ
(a)(b)Figure 4
Comparison ofABC4Γ and GA5Γ


## 4. Results for Blue Starch-Iodine Complex

The main structure of amylose are cyclic degradants known as cyclodextrins. They are obtained enzymatically and may be considered as single turns of the helix of amylose imploding into a circular path. In all of these complexes, cyclodextrin molecules are positioned in front to form dimers and they are piled together to generate large cylinders that resemble the amylose helix in its global structure. The most interesting one is (trimesic acidH20)H10I5 with linear polyiodide chain. This structural model was accepted, but, unfortunately, cannot shed light on the actual configuration of the polyiodide chain (see Figure 5 for the molecular structure of blue starch-iodine and Figure 6 for its unit graph and the graph model corresponding to blue starch-iodine for n = 6, where n is the number of units). In starch-iodine, there are three types of vertices having degrees 1, 2, and 3. For n≥3, blue starch-iodine complex has five types of edge partitions as(26)E1,2Γ=ℜu˜=1,ℜv˜=2andu,v∈VΓ,E1,3Γ=ℜu˜=1,ℜv˜=3andu,v∈VΓ,E2,2Γ=ℜu˜=2,ℜv˜=2andu,v∈VΓ,E2,3Γ=ℜu˜=2,ℜv˜=3andu,v∈VΓ,E3,3Γ=ℜu˜=3,ℜv˜=3andu,v∈VΓ.Figure 5
Molecular structure of blue starch-iodine.Figure 6
(a) Graph of blue starch-iodine forn = 1 and (b) graph of blue starch-iodine for n = 6.Theorem 4.
For alln≥3, let Γ be the graph of blue starch-iodine complex, we have the following RαΓ,α∈R:(27)R1Γ=39n2−n+12,R1/2Γ=16.429n2−1.0354+2.8695,R−1/2Γ=3.0272n2+.5585n−0.6764,R−1Γ=0.75n2+0.6944n−0.6667.Proof.
Forα = 1, the general Randić connectivity index is(28)R1Γ=∑uv∈EΓℜu˜ℜv˜.
From Table5 (for edge partition) and equation (2), we get(29)R1Γ=2n1×2+∑i=1n−1n+2−21×3+n2×2+∑i=1n−16n−2+22×3+4n3×3=39n2−n+12.
Now, forα=1/2, we have(30)R1/2Γ=∑uv∈EΓℜu˜ℜv˜.
By using Table5 and equation (2), after simplification, we have(31)R1/2Γ=2n1×2+∑i=1n−1n+2−21×3+n2×2+∑i=1n−16n−2+22×3+4n3×3=16.429n2−1.0354+2.8695.
Forα=−1/2, we have(32)R−1/2Γ=∑uv∈EΓ1ℜu˜ℜv˜.
From Table5 and equation (2), it follows that(33)R−1/2Γ=2n1×2+∑i=1n−1n+2−21×3+n2×2+∑i=1n−16n−2+22×3+4n3×3=3.0272n2+.5585n−0.6764.
Forα=−1, we have(34)R−1Γ=∑uv∈EΓ1ℜu˜ℜv˜.
Again by using Table5 and equation (2), we get(35)R−1Γ=2n1×2+∑i=1n−1n+2−21×3+n2×2+∑i=1n−16n−2+22×3+4n3×3=0.75n2+0.6944n−0.6667.Table 5
Edge partition based on the degree of vertices.
Types of edgesE1,2E1,3E2,2E2,3E3,3Number of edges(1, 2)(1, 3)(2, 2)(2, 3)(3, 3)Frequency2n∑i=1n−1n+2−2n∑i=1n−16n−2+24nTheorem 5.
For alln≥3, let Γ be the graph of blue starch-iodine complex, then we have the following:(36)M1Γ=34n2−2n+4,M2Γ=39n2−n+12,ABCΓ=5.0591n2−0.0523n−0.4376,GAΓ=6.7448n2−3.0868n+0.4552.Proof.
By using Table5 and equation (4), we get(37)M1Γ=∑uv∈EΓℜu˜+ℜv˜=2n1+2+∑i=1n−1n+2−21+3+n2+2+∑i=1n−16n−2+22+3+4n3+3=34n2−2n+4,M2Γ=∑uv∈EΓℜu˜ℜv˜=2n1×2+∑i=1n−1n+2−21×3+n2×2+∑i=1n−16n−2+22×3+4n3×3=39n2−n+12.
By using Table5 and equation (5), we get(38)ABCΓ=∑uv∈EΓℜu˜+ℜv˜−2ℜu˜ℜv˜=2n1+2−21×2+∑i=1n−1n+2−21+3−21×3+n2+2−22×2+∑i=1n−16n−2+22+3−22×3+4n3+3−23×3=5.0591n2−0.0523n−0.4376.
By using Table5 and equation (6), we get(39)GAΓ=∑uv∈EΓ2ℜu˜ℜv˜ℜu˜+ℜv˜=22n1×21+2+∑i=1n−1n+2−21×31+3+n2×22+2+∑i=1n−16n−2+22×32+3+4n3×33+3=6.7448n2−3.0868n+0.4552.Theorem 6.
For alln≥3, let Γ be the graph of blue starch-iodine complex, we have(40)ABC4Γ=4.0798n2−0.7682n+0.04,GA5Γ=7.8987n2−3.1339n+1.1727.Proof.
By using Table6 and equation (7), we get(41)ABC4Γ=∑uv∈EΓSu+Sv−2SuSv=n2+3−22×3+n2+4−22×4+n3+5−23×5+n−23+6−23×6+n2−n−13+7−23×7+n4+8−24×8+n5+7−25×7+n−26+6−26×6+3n2−5n+46+7−26×7+n6+8−26×8+2n−47+7−27×7+4n2−7n+67+8−27×8=4.0798n2−0.7682n+0.04.
By using Table6 and equation (8), we get(42)GA5Γ=∑uv∈EΓ2SuSvSu+Sv=2n2×32+3+n2×42+4+n3×53+5+n−23×63+6+n2−n−13×73+7+n4×84+8+n5×75+7+n−26×66+6+3n2−5n+46×76+7+n6×86+8+2n−47×77+7+4n2−7n+67×87+8=7.8987n2−3.1339n+1.1727.Table 6
Edge partition based on the degree sum of end vertices of each edge.
Types of edgesNo. of edgesFrequencyE2,3(2, 3)nE2,4(2, 4)nE3,5(3, 5)nE3,6(3, 6)∑i=1n−11−1E3,7(3, 7)∑i=1n−1n−1E4,8(4, 8)nE5,7(5, 7)nE6,6(6, 6)∑i=1n−11−1E6,7(6, 7)∑i=1n−13n−2+2E6,8(6, 8)nE7,7(7, 7)∑i=1n−12−2E7,8(7, 8)∑i=1n−14n−3+3


## 5. Numerical and Graphical Representation

Here, we give numeric and graphic representation for the results calculated in the above section (see Tables7 and 8 ).Table 7
Numerical comparison ofM1Γ, M2Γ, ABCΓ, GAΓ, R1Γ, R−1Γ, R1/2Γ, and R−1/2Γ.
nM1ΓM2ΓABCΓGAΓR1ΓR−1ΓR1/2ΓR−1/2Γ136504.56924.1132501.36118.26332.9093213616619.694221.26081666.055366.514812.5488330436044.937451.89836013.4162147.624328.2419454063280.298896.024863223.4437261.591849.98875844982125.7784153.641298236.1378408.417377.7891612161410181.3762224.7472141051.4985588.1007111.6433716561916247.0922309.3428191669.5258800.6421151.5512821642500322.9264407.428250090.21971046.0415197.5126927403162408.8788519.00283162113.58021324.2989249.52781033843902504.9494644.06723902139.60731635.4143307.5967Table 8
Numerical comparison ofABC4Γ and GA5Γ.
n12345678910ABC4Γ3.3514.8234.4562.2498.19142.30194.57255323.59400.34GA5Γ5.9426.4962.86115.02182.97266.72366.27481.62612.76759.70


## 6. Conclusion

Amylose has a significant function in the storage of plant energy. It is not easy to digest compared to amylopectin; however, it occupies less space than amylopectin due to its spiral structure. Consequently, for storage in plants, it is the preferred starch. A mixture of iodine and potassium iodide in water is light orange-brown. When added to a sample containing starch, such as the bread pictured above, the colour will change to a deep blue (see the comparison of different indices in Figures7 and 8 ). In this study, we have calculated degree-dependent topological indices of amylose and blue starch-iodine. We observed that R−1/2 is closely related to geometric-arithmetic, R−1 is closely related to atom-bond connectivity and modified atom-bond connectivity, the second Zagreb is the first Randic index, while R1/2 is approximately equal to the modified geometric-arithmetic of amylose. Similarly, other observations can take place from the graphical representations given in this paper.Figure 7
(a) Comparison ofRαΓ for 1, −1, 1/2, −1/2 and (b) comparison of M1Γ, M2Γ, ABCΓ, and GAΓ.
(a)(b)Figure 8
Comparison ofABC4Γ and GA5Γ.



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