Anam Rani
email: arani@kfu.edu.sa

Usman Ali
email: uali@bzu.edu.pk

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

Keywords: Polysaccharides, Amylose, Blue Starch Iodine Complex, Zagreb index, Randić index, Atombond connectivity index, Geometric arithmetic index

Starch is a polymer of glucose where alpha-linkages 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 index M 1 (Γ) and M 2 (Γ), Randić index R α (Γ) for α = -1 2 , -1, 1 2 , 1, Atom-bond connectivity index ABC(Γ), Geometric arithmetic index GA(Γ), fourth Atombond connectivity index ABC 4 (Γ) and fifth Geometric arithmetic index GA 5 (Γ), degree-based topological indices a graph Γ representing Polysaccharides, namely, Amylose and Blue Starch Iodine Complex. In the end, we compare these indices and depict their graphic behavior.

Introduction

Amylose has the most basic structure of all nutritional polysaccharides, composed purely of glucose polymers connected only by α(1 -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 is 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 [START_REF] Deutsch | M-polynomial and degree-based topological indices[END_REF] described the M-polynomial and degree-based topological indices of graphs. The articles [START_REF] Gupta | On the symmetric division deg index of graph[END_REF][START_REF] Gutman | Topological indices[END_REF] discussed the symmetric divisor deg index of graphs, first Zegrab after 30 years in changed form and topological indices of molecular structure. The authors in [START_REF] Gutman | Graph theory and molecular orbitals. total ϕ-electron energy of alternant hydrocarbons[END_REF] also discussed the π electron energy of hydrocarbons. In the recent years, Hasni et al. computed the degree based topological indices of line graph of benzene ring embedded in P-type-surface in 2D network [START_REF] Ahmad | Computing the degree based topological indices of line graph of benzene ring embedded in P-type-surface in 2D network[END_REF]. In [START_REF] Nadeem | On the edge version of geomeetric-arithmetic index of nanocones[END_REF] the authors calculated the index numbers for the edge version of geomeetric-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.

Let Γ be connected simple graph with V (Γ) a set of vertices and E(Γ) a set of edges. Let u ∈ V (Γ) and its degree is represented by ℜ u . The idea of degree-based topological indices began from Wiener index, in 1945, Wiener defined them while studying alkane's boiling point cf. [START_REF] Wiener | Structural determination of paraffin boiling points[END_REF]. The first degree-based topological index is Randić index given by Milan Randić in [START_REF] Randić | Characterization of molecular branching[END_REF] and is described as:

R -1 2 = ∑ uv∈E(Γ) 1 ℜ u ℜ v Generalized Randić index (denoted as R α (Γ) ) is described as: R α (Γ) = ∑ uv∈E(Γ) ( ℜ u ℜ v ) α α = 1, 1 2 , - 1 2 , -1 (1)
Inverse generalized Randić index (denoted as RR α (Γ)) is described as:

RR α (Γ) = ∑ uv∈E(Γ) 1 ( ℜ u ℜ v ) α
In [START_REF] Gutman | The first zagreb index 30 years after[END_REF][START_REF] Gutman | Graph theory and molecular orbitals. total ϕ-electron energy of alternant hydrocarbons[END_REF] and [START_REF] Trinajstić | About the zagreb indices, Kemija u industriji[END_REF], Gutman and Trinajstić introduced and defined the first Zagreb index (denoted as M 1 (Γ)) and second Zagreb index (denoted as M 2 (Γ)) as:

M 1 (Γ) = ∑ uv∈E(Γ) ( ℜ u + ℜ v ) M 2 (Γ) = ∑ uv∈E(Γ) ( ℜ u ℜ v ) (2)
In [START_REF] Estrada | An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes[END_REF], Estrada introduced and studied about the Atom-bond connectivity index (denoted as ABC(Γ)). It is defined as:

ABC(Γ) = ∑ uv∈E(Γ) ℜ u + ℜ v -2 ℜ u ℜ v (3)
Geometric-arithmetic index (denoted as GA(Γ)) was given by Vuki čevi ć cf. [START_REF] Vuki | Topological index based on the ratios of geometrical and arithmetical means of end-vertex degree of edges[END_REF] and is defined as:

GA(Γ) = ∑ uv∈E(Γ) 2 ℜ u ℜ v ℜ u + ℜ v (4)
The fourth version of the ABC index (denoted as ABC 4 (Γ)) was introduced by Ghorbani in [START_REF] Ghorbani | Computing ABC 4 index of nanostar dendrimers[END_REF] and is defined as:

ABC 4 (Γ) = ∑ uv∈E(Γ) S u + S v -2 S u S v (5)
where

S u = ∑ v=N Γ (u) ℜ v and N Γ (u) = {v ∈ V (Γ)|uv ∈ E(Γ)}
The fifth version of the GA index (denoted as GA 5 (Γ)) was given by Graovac cf. [START_REF] Graovac | Computing fifth geometric-arithmetic index for nanostar dendrimers[END_REF] and is defined as:

GA 5 (Γ) = ∑ uv∈E(Γ) 2 √ S u S v S u + S v (6)
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 of α-D-glucose units, linked by α(1 -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 [START_REF] Green | Which starch fraction is water-soluble, Amylose or Amylopectin?[END_REF] (see Figure1 for a molecular structure of Amylose and Figure2 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:

E 1,2 (Γ) = { ℜ u = 1, ℜ v = 2 and u, v ∈ V (Γ)} E 1,3 (Γ) = { ℜ u = 1, ℜ v = 3 and u, v ∈ V (Γ)} E 2,3 (Γ) = { ℜ u = 2, ℜ v = 3 and u, v ∈ V (Γ)} E 3,3 (Γ) = { ℜ u = 3, ℜ v = 3 and u, v ∈ V (Γ)} Figure 1: Molecular structure of Amylose Types of edges E {1,2} E {1,3} E {2,3} E {3,3} Edges (1,2) (1,3) (2,3) (3,3) Frequency n 2n + 2 5n -2 4n
Table 1: Edge partition of edges based on degree of vertices Theorem 1. For all n ≥ 2, the graph Γ of Amylose, we have the following:

R 1 (Γ) = 74n -6 R1 2 (Γ) = 29.1258 n -1.4349 R -1 2 (Γ) = 5.2363 n + 0.3382 R -1 (Γ) = 2.4444 n + 0.3334 Proof. The general Randić connectivity index R α (Γ) for α = 1 is R 1 (Γ) = ∑ uv∈E(Γ) ℜ u ℜ v
From Table 1 and Equation 1, we get

R 1 (Γ) = n(1 × 2) + (2n + 2)(1 × 3) + (5n -2)(2 × 3) + 4n(3 × 3) = 74n -6 Now, for α = 1 2 , the general Randić connectivity index R α (Γ) is R1 2 (Γ) = ∑ uv∈E(Γ) ℜ u ℜ v
Again, from Table 1 and Equation 1, we have

R1 2 (Γ) = n (1 × 2) + (2n + 2) (1 × 3) + (5n -2) (2 × 3) + 4n (3 × 3) = 29.1258 n -1.4349 If α = -1 2 , then R -1 2 (Γ) = ∑ uv∈E(Γ) 1 ℜ u ℜ v
From Table 1 and Equation 1, it follows that

R -1 2 (Γ) = n (1 × 2) + (2n + 2) (1 × 3) + (5n -2) (2 × 3) + 4n (3 × 3) = 5.2363 n + 0.3382 Now, for α = -1, we have R -1 (Γ) = ∑ uv∈E(Γ) 1 ℜ u ℜ v
From Table 1 and Equation 1, we get

R -1 (Γ) = n (1 × 2) + (2n + 2) (1 × 3) + (5n -2) (2 × 3) + 4n (3 × 3) = 2.4444 n + 0.3334
Theorem 2. For all n ≥ 2, Γ be the graph of Amylose. Then we have the following:

M 1 (Γ) = 60n -2 M 2 (Γ) = 74n -6 ABC(Γ) = 8.5423n + 0.2188 GA(Γ) = 11.5738n -0.2276
Proof. By using Table 1 and Equation 2, we get

M 1 (Γ) = ∑ uv∈E(Γ) ( ℜ u + ℜ v ) = n(1 + 2) + (2n + 2)(1 + 3) + (5n -2)(2 + 3) + 4n(3 + 3) = 60n -2 M 2 (Γ) = ∑ uv∈E(Γ) ℜ u ℜ v = n(1 × 2) + (2n + 2)(1 × 3) + (5n -2)(2 × 3) + 4n(3 × 3) = 74n -6
By using Table 1 and Equation 3, we get

ABC(Γ) = ∑ uv∈E(Γ) ℜ u + ℜ v -2 ℜ u ℜ v = n 1 + 2 -2 1 × 2 + (2n + 2) 1 + 3 -2 1 × 3 + (5n -2) 2 + 3 -2 2 × 3 + 4n 3 + 3 -2 3 × 3 = 8.5423n + 0.2188
By using Table 1 and Equation 4, we get

GA(Γ) = ∑ uv∈E(Γ) 2 ℜ u ℜ v ℜ u + ℜ v = 2 n √ 1 × 2 (1 + 2) + (2n + 2) √ 1 × 3 (1 + 3) + (5n -2) √ 2 × 3 (2 + 3) + 4n √ 3 × 3 (3 + 3) = 11.5738n -0.2276
In the following table, we give the edge partition centered on degree sum of end vertices for each edge.

Types of edges E

{2,4} E {3,6} E {3,7} E {4,7} E {6,6} E {6,7} E {6,8} E {7,7} E {7,8}
GA 5 (Γ) = ∑ uv∈E(Γ) 2 √ S u S v S u + S v = 2 n √ 2 × 4 (2 + 4) + √ 3 × 6 (3 + 6) + (2n + 1) √ 3 × 7 (3 + 7) + n √ 4 × 7 (4 + 7) + √ 6 × 6 (6 + 6) + (3n -1) √ 6 × 7 (6 + 7) + (n -1) √ 6 × 8 (6 + 8) + (2n + 1) √ 7 × 7 (7 + 7) + (2n -2) √ 7 × 8 (7 + 8) = 11.7142n -0.123 Table 3: Numerical Comparison of M 1 (G), M 2 (Γ), ABC(Γ), GA(Γ), R 1 (Γ), R -1 (Γ), R1 2 (Γ), and R -1 2 (Γ) n M 1 (Γ) M 2 (Γ) ABC(Γ) GA(Γ) R 1 (Γ) R -1 (Γ) R1 2 (Γ) R -1 2 (Γ)

Numerical and Graphical Representation

" The numeric representation of the results calculated above is illustrated in Tables 3 and4, while the graphic representation is devoted to Figures 3 and4." The main structure for 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

E 1,2 (Γ) = { ℜ u = 1, ℜ v = 2 and u, v ∈ V (Γ)} E 1,3 (Γ) = { ℜ u = 1, ℜ v = 3 and u, v ∈ V (Γ)} E 2,2 (Γ) = { ℜ u = 2, ℜ v = 2 and u, v ∈ V (Γ)} E 2,3 (Γ) = { ℜ u = 2, ℜ v = 3 and u, v ∈ V (Γ)} E 3,3 (Γ) = { ℜ u = 3, ℜ v = 3 and u, v ∈ V (Γ)}
Types of edges E {1,2} E {1,3} E {2,2} E {2,3} E {3,3} Number of edges (1,2) (1,3) (2,2) (2,3) (3,3) Frequency 2n ∑ n-1 i=1 (n + 2) -2 n ∑ n-1 i=1 (6n -2) + 2 4n
Table 5: Edge partition based on degree of vertices Theorem 4. For all n ≥ 3, the graph Γ of blue starch-iodine complex, we have the following R α (Γ), α ∈ R :

R 1 (Γ) = 39n 2 -n + 12 R1 2 (Γ) = 16.429n 2 -1.0354 + 2.8695 R -1 2 (Γ) = 3.0272n 2 + .5585 n -0.6764 R -1 (Γ) = 0.75n 2 + 0.6944n -0.6667
Proof. For α=1, the general Randić connectivity index is

R 1 (Γ) = ∑ uv∈E(Γ) ℜ u ℜ v
From Table 5 (for edge partition) and Equation 1, we get

R 1 (Γ) = 2n(1 × 2) + [ n-1 ∑ i=1 (n + 2) -2](1 × 3) + n(2 × 2) + [ n-1 ∑ i=1 (6n -2) + 2](2 × 3) + 4n(3 × 3) = 39n 2 -n + 12
Now, for α = 1 2 , we have

R1 2 (Γ) = ∑ uv∈E(Γ) ℜ u ℜ v
By using Table 5 and Equation 1, after simplification, we have

R1 2 (Γ) = 2n (1 × 2) + [ n-1 ∑ i=1 (n + 2) -2] (1 × 3) + n (2 × 2) + [ n-1 ∑ i=1 (6n -2) + 2] (2 × 3) + 4n (3 × 3) = 16.429n 2 -1.0354 + 2.8695
For α = -1 2 , we have

R -1 2 (Γ) = ∑ uv∈E(Γ) 1 ℜ u ℜ v
From Table 5 and Equation 1, it follows that

R -1 2 (Γ) = 2n (1 × 2) + [∑ n-1 i=1 (n + 2) -2] (1 × 3) + n (2 × 2) + [∑ n-1 i=1 (6n -2) + 2] (2 × 3) + 4n (3 × 3) = 3.0272n 2 + .5585 n -0.6764
For α = -1, we have

R -1 (Γ) = ∑ uv∈E(Γ) 1 ℜ u ℜ v
Again by using Table 5 and Equation 1, we get

R -1 (Γ) = 2n (1 × 2) + [∑ n-1 i=1 (n + 2) -2] (1 × 3) + n (2 × 2) + [∑ n-1 i=1 (6n -2) + 2] (2 × 3) + 4n (3 × 3) = 0.75n 2 + 0.6944n -0.6667
Theorem 5. For all n ≥ 3, Γ be the graph of blue starch-iodine complex. Then we have the following:

M 1 (Γ) = 34n 2 -2n + 4 M 2 (Γ) = 39n 2 -n + 12
ABC(Γ) = 5.0591n 2 -0.0523n -0.4376 GA(Γ) = 6.7448n 2 -3.0868n + 0.4552 Proof. By using Table 5 and Equation 2, we get

M 1 (Γ) = ∑ uv∈E(Γ) ( ℜ u + ℜ v ) = 2n(1 + 2) + [ n-1 ∑ i=1 (n + 2) -2](1 + 3) + n(2 + 2) + [ n-1 ∑ i=1 (6n -2) + 2](2 + 3) + 4n(3 + 3) = 34n 2 -2n + 4 M 2 (Γ) = ∑ uv∈E(Γ) ℜ u ℜ v = 2n(1 × 2) + [ n-1 ∑ i=1 (n + 2) -2](1 × 3) + n(2 × 2) + [ n-1 ∑ i=1 (6n -2) + 2](2 × 3) + 4n(3 × 3) = 39n 2 -n + 12
By using Table 5 and Equation 3, we get

ABC(Γ) = ∑ uv∈E(Γ) ℜ u + ℜ v -2 ℜ u ℜ v = 2n 1 + 2 -2 1 × 2 + [ n-1 ∑ i=1 (n + 2) -2] 1 + 3 -2 1 × 3 + n 2 + 2 -2 2 × 2 + [ n-1 ∑ i=1 (6n -2) + 2] 2 + 3 -2 2 × 3 + 4n 3 + 3 -2 3 × 3 = 5.0591n 2 -0.0523n -0.4376
By using Table 5 and Equation 4, we get

GA(Γ) = ∑ uv∈E(Γ) 2 ℜ u ℜ v ( ℜ u + ℜ v ) = 2 2n √ 1 × 2 (1 + 2) + [ n-1 ∑ i=1 (n + 2) -2] √ 1 × 3 (1 + 3) + n √ 2 × 2 (2 + 2) + [ n-1 ∑ i=1 (6n -2) + 2] √ 2 × 3 (2 + 3) + 4n √ 3 × 3 (3 + 3) = 6.7448n 2 -3.0868n + 0.4552 Types of edges E {2,3} E {2,4} E {3,5} E {3,6} E {3,7} E {4,8} No. of edges (2,3) (2,4) (3,5) (3,6) (3,7) (4,8) Frequency n
n n ∑ n-1 i=1 (1) -1 ∑ n-1 i=1 (n) -1 n Types of edges E {5,7} E {6,6} E {6,7} E {6,8} E {7,7} E
n ∑ n-1 i=1 (1) -1 ∑ n-1 i=1 (3n -2) + 2 n ∑ n-1 i=1 (2) -2 ∑ n-1 i=1 (4n -3) + 3
Table 6: Edge partition based on degree sum of end vertices of each edge Theorem 6. For all n ≥ 3, the graph Γ of blue starch-iodine complex, we have ABC 4 (Γ) = 4.0798n 2 -0.7682n + 0.04

GA 5 (Γ) = 7.8987n 2 -3.1339n + 1.1727
Proof. By using Table 6 and Equation 5, we get

ABC 4 (Γ) = ∑ uv∈E(Γ) S u + S v -2 S u S v = n 2 + 3 -2 2 × 3 + n 2 + 4 -2 2 × 4 + n 3 + 5 -2 3 × 5 + (n -2) 3 + 6 -2 3 × 6 + (n 2 -n -1) 3 + 7 -2 3 × 7 + n 4

Conclusion

Amylose has a significant function in the storage of plant energy. It is not easy to digest 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 color will change to a deep blue (see the comparison of different indices in Figures 7 and8 ). 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 bond and modified atom bond connectivity, the second zegrab 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 2 :

 2 Figure 2: (a) Graph of Amylose for n=1 (b) Graph of Amylose for n=4

Figure 3 :

 3 Figure 3: (a) Comparison of R α for α = 1, -1, 1 2 , -1 2 , (b) Comparison of M 1 (Γ), M 2 (Γ), ABC(Γ) and GA(Γ)

Figure 4 :

 4 Figure 4: Comparison of ABC 4 (Γ), GA 5 (Γ)

Figure 5 :Figure 6 :

 56 Figure 5: (Molecular structure of Blue Starch-Iodine)

Figure 7 :Figure 8 :

 78 Figure 7: (a) Comparison of R α (Γ) for 1, -1, 1 2 , -1 2 , (b) Comparison of M 1 (Γ), M 2 (Γ), ABC(Γ) and GA(Γ)

Table 2 :

 2 Edge partition based on degree sum of end vertices of each edge

	Number of edges (2,4) (3,6)		(3,7)	(4,7) (6,6)	(6,7)	(6,8)	(7,7)	(7,8)
	Frequency	n		1	2n + 1	n	1	3n -1 n -1 2n + 1 2n -2
			S u + S v -2
	uv∈E(Γ)		S u S v
	= n	2 + 4 -2 2 × 4	+	3 + 6 -2 3 × 6	+ (2n + 1)	3 + 7 -2 3 × 7	+ n	4 + 7 -2 4 × 7	+	6 + 6 -2 6 × 6
	+ (3n -1)	6 + 7 -2 6 × 7	+ (n -1)	6 + 8 -2 6 × 8	+ (2n + 1)	7 + 7 -2 7 × 7	+ (2n -2)	7 + 8 -2 7 × 8
	= 6.4972n + 0.2874
	By using Table 2 and Equation 6, we get

Theorem 3. For all n ≥ 2, the graph Γ of Amylose, we have ABC 4 (Γ) = 6.4972n + 0.2874 GA 5 (Γ) = 11.7142n -0.123 Proof. By using Table

2

and Equation 5, we get ABC 4 (Γ) = ∑

Table 4 :

 4 Numerical Comparison of ABC 4 (Γ) and GA 5 (Γ)

	1	58	68	8.7611 11.3462	68	2.7778	27.6909 5.5746
	2	118	142	17.3034	22.92		142	5.2222	56.8166 10.8109
	3	178	216	25.8457 34.4938	216	7.6667	85.9424 16.0474
	4	238	290	34.388	46.0676	290 10.1111 115.0682 21.2837
	5	298	364	42.9303 57.6414	364 12.5556 144.1939 26.5201
	6	358	438	51.4726 69.2152	438	15	173.3197 31.7565
	7	418	512	60.0149 80.789	512 17.4444 202.4455 36.9929
	8	478	586	68.5572 92.3628	586 19.8889 231.5712 42.2293
	9	538	660	77.0995 103.9366 660 22.3333 260.6969 47.4656
	10	598	734	85.6418 115.5104 734 24.7778 289.8228 52.7020
	n	1	2	3	4	5	6	7	8	9	10
	ABC 4 (Γ) 6.785 13.282 19.779 26.276 32.773 39.271 45.768 52.265	58.762	65.259
	GA 5 (Γ) 11.591 23.305 35.019 46.734 58.448 70.162 81.876 93.5906 105.305 117.019

+ (3n

Numerical and Graphical Representation

Here, we give numeric and graphic representation for the results calculated in the above section (see Table 7 and8).