# On Computation of Entropy Measures and Molecular Descriptors for Isomeric Natural Polymers

Authors: Shazia Manzoor; Muhammad Kamran Siddiqui; Sarfraz Ahmad; Samuel Asefa Fufa
Journal: Journal of Mathematics
 (2022)

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## Abstract

Glycogen is a polysaccharide that has a large number of highly branched polymers. It has a structure that is nearly identical to that of amylopectin. It can be found in practically all animal cells and some plant cells. Glycogen is a natural polysaccharide polymer with features that make it a good antiparticle carrier for cancer therapeutics. It is not only biocompatible by nature but also be chemically modified to accommodate additional molecular components. Topological indices are used to create quantitative structure-activity relationships (QSARs), in which the biological activity or other properties of molecules are linked to their chemical structure. We estimated certainK^ Banhatti and Gourava indices of natural polymers of polysaccharides, namely, glycogen and amylopectin, which have therapeutic applications, extraordinary features, and fascinating molecular framework, in this study. We also discovered some relationships between K^ Banhatti indices and information entropies, as well as a relationship between Gourava indices and their respective information entropies. In addition, we give a comparative analysis of these macromolecule families using graphs to highlight their nature.


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## Body

## 1. Introduction

A polymer is a material made up of many repetitive subunits and consists of very massive molecules. Both synthetic and natural polymers perform crucial and pervasive character in daily occurrences due to their extensive range of characteristics. Polymers serve an important role in medication administration and prosthodontics materials, and their popularity has been phenomenal. Drug delivery and prosthodontics substances both rely on polymers, which have phenomenal popularity (Figure1). Polymeric networks have features that are determined not only by their chemical framework but also by how isomer chains are connected together to form a network [1]. With regarded to their molecular chains, polymers are classified into four major categories.Figure 1
Depiction of polymers in terms of structural chains. (a) Linear polymer. (b) Branched polymer. (c) Cross-linked polymer. (d) Network polymer.
(a)(b)(c)(d)Macromolecules (class of polymers) are found in practically every dietary item. Polysaccharides (branched or linear) such as starch and cellulose are found in the majority of foods. These polysaccharides have two basic biological functions: energy storage for metabolism and to provide structural support (cellulose). Cellulose is made out of the same substance as graphite and diamonds, but it has a different structure. The chemical formula of cellulose isC6H10O5n. It is a natural polymer of glucose. Monomers (monosaccharides) are tiny pieces that make up natural polymers, and glucose is the most fundamental component of cellulose [2]. Carbon, hydrogen, and oxygen are found in glucose, a form of sugar. These components combine to form a hexagonal structure with six carbon atoms and one carbon atom protruding from the end. To create many types of frameworks, distinct glucose rings can be connected at various carbons. Some ring segments are reversed, resulting in two separate types of glucose named as α-glucose and β-glucose. The alcohol OH linked to C1 is downwards in α-glucose, while alcohol OH linked to C1 is upwards in β-glucose. In amylopectin and glycogen, there are two types of glycosidic bonding: αC1−C4 and βC1−C6 glycosidic bonding. Natural polymers, especially those derived from carbohydrates, have been proven to have a wide range of pharmaceutical applications [3–5], as shown in Figure 2.Figure 2
Fundamental unit ofα-glucose and β-glucose with their connections.Mathematical chemistry has a section called chemical graph theory. It has a significant impact on the progress of chemical sciences. Physicochemical features of chemical substances are frequently characterized in chemical science using molecular-based structural descriptors, also known as topological indices. Vertices represent atoms, while edges represent bonds in a molecular graph. A topological index for a subatomic graph is an individual quantity that can be employed to characterize some aspects of the graph. In QSPR/QSAR research, a number of topological indices have been used [6–9]. A chemical graph H=VH,EH is an ordered pair of two finite sets VH and EH, where VH is the set of vertices (atoms) and EH is the set of edges (bonds) in chemical graph H. The valence of molecules is usually portrayed by the vertex degrees [10, 11].Ifθ=bw is an edge of graph H, then the vertex b and edge θ are incident as well as w and θ. Let ψθ represents the degree of an edge θ=bw in H, which is computed as ψθ=ψb+ψw−2. The multi-Banhatti indices are defined, as given in Tables 1 and 2, respectively:Table 1
Topological descriptors.
Banhatti topological descriptorsExpressionFirstK^ Banhatti index Bλ1H1, where λ=1 [12, 13]∑γμ∈EH1ψγ+ψμ+ψγ+ψμSecondK^ Banhatti index B2λH1, where ω=1 [12]∑γμ∈EH1ψγ+ψμ+ψγ+ψμHarmonic Banhatti index,hBH [14]∑βμ2/ψβ+ψμSum connectivity Banhatti index,SB1−12H1 [15, 16]∑γμ∈EH11/ψγ+ψθ+1/ψμ+ψθFirstK^-hyper Banhatti index, HB12H1 [17, 18]∑γμ∈EH1ψγ2+ψμ2+ψγ2+ψμ2SecondK^-hyper Banhatti index, HB22H1 [4, 17]∑γμ∈EH1ψγ2×ψμ2+ψγ2×ψμ2Modified first Banhatti index,mB1−1H1 [14]∑γμ∈EH1ψγ−1+ψμ−1+ψγ−1+ψμ−1Modified second Banhatti index,mB2−1H1 [14]∑γμ∈EH1ψγ−1×ψμ−1+ψγ−1×ψμ−1Table 2
Gourava topological descriptors.
Gourava topological descriptorsExpressionFirst Gourava index,GV1H1 [19]∑γμ∈EH1ψγ+ψμ+ψγ+ψμSecond Gourava index,GV2H1 [19]∑γμ∈EH1ψγ+ψμ+ψγ+ψμProduct connectivity Gourava index,SGVH1 [20]∑γμ∈EH11/ψγ+ψμ×ψγψμSum connectivity Gourava index,SGVH1 [21]∑γμ∈EH11/ψγ+ψμ+ψγ×ψμFirst hyper-Gourava index,GVh1H1 [22]∑γμ∈EH1ψγ+ψμ+ψγ×ψμSecond hyper-Gourava index,GVh2H1 [23]∑γμ∈EH1ψγ+ψμ×ψγψμIn chemistry, information entropy [24, 25] is now used in two modes. First, it is a structural descriptor for assessing the complexity of chemical structures [26]. Information entropy is useful in this regard for connecting structural and physicochemical features [27], numerically distinguishing isomers of organic molecules [28], and classifying natural products and synthetic chemicals [29, 30]. The physicochemical sounding of information entropy is a different mode of application. As a result, Kobozev demonstrated its utility in analysing physicochemical processes that simulate information transmission [31]. Manzoor et al. [32] used entropy values to study organic compound chemical processes. The information entropy [33] based on Shannon’s entropy [34] is described as(1)EH=log2Γa−1Γa∑i=1jFiΔbiwilog2Δbiwi,where Γa=∑i=1jFiΔbiwi is used for the topological indices, j is the number of edge types, Fi denotes the frequency of repetition, and Δbiwi is the weight of edge bw in the graph.


## 2. Formation of GlycogenGLμγ Network Planar Graph

Glycogen is a polycarbohydrate of glucose (form of energy storage in the body). Glycogen is generated and stored mostly in the liver and skeletal muscle cells of humans [35]. Other organs and cells, such as the kidneys, corpuscles, leucocytes, and neuroglia in the brain, contain small amounts of glycogen. Claude Bernard (French scientist) discovered glycogen, which is found mostly in the liver and muscles [36]. Natural polymers including cellulose, proteins, chitin, carbs, and glycogen are excellent sources of energy since they are essential for life to continue. The experiential formula for glycogen is C6H10O5n. Glycogen is a massive, complicated, and branching polymer made up of around 30,000 glucose monomers. It is made up of chains of glucose molecules connected linearly by αC1−C4 glycosidic connections. A chain of glucose split up after every ten to twelve residues via αC1−C6 glycosidic connections. In glycogen, this type of bonding results in forking and rambling patterns. On the other hand, cellulose (a natural polymer) contains βC1−C4 glycosidic connections, resulting in a more rigid linear chain. As a result, cellulose are unable to be broken down in our stomachs. There is only one diminishing end to glycogen; however, there are many nonreductive ends. It is shown in Figure 3.Figure 3
Molecular structure of the glycogenGLμγ network.In human and animal cells, glycogen molecules comprehend glucose as the primary storage reservoir. Glycogen is easily digested to release glucose when needed. Even if the supply of glucose in blood is unequal, this self-regulating process keeps the amount of glucose in blood at a consistent level. The Cori cycle (Figure4) revolves around the interplay between glycogen and glucose.Figure 4
The Cori cycle.Despite the fact that enough research has been done on the role of hormones like glucagon, adrenaline, and insulin in the control of the glycogen metabolism, it is still a significant issue of research [37–39]. αC1−C6 glycosidic bonding takes place in glycogen after every 10 glucose residues, forming the branch. The glycogen GLμγ network is fabricated in μ−1 ramifications of length 0≤γ≤1 and is created from the glycogen molecule, as shown in Figure 5.Figure 5
Hydrogen-diminished molecular graph of the glycogenGLμγ network.LetVGLμγ and EGLμγ represent the vertex set and the edge set of GLμγ, in which there are three types of vertices and four types of edges. The vertices having degrees 1, 2, and 3 and by using fundamental counting rule, ∣V1∣=γμ−1+9μ, ∣V2∣=22γμ−1+21μ−1, and ∣V3∣=3γμ−1+29μ−2. The edge partition is grounded on the degrees of terminal vertices of every edge, as given in Table 3. By using the fundamental counting technique, ∣E1∣=γμ−1+9μ, ∣E2∣=γμ−1+12μ, ∣E3∣=6γμ−1+60μ−4, and ∣E4∣=γμ−1+9μ−1. After some basic computation, we can see that ∣VGLμγ∣=8γμ−1+80μ−4 and ∣EGLμγ∣=9γμ−1+90μ−5. The principle strategy utilized here is the way to deal with edge partitioning and vertices degree calculating. Subsequently, in [40], many degree-based topological indices were calculated. We computed some K^ Banhatti indices with corresponding K^ Banhatti entropies. To the best of our knowledge, no such work has ever been done before.Table 3
Edge description ofGLμγ.
ψγ,ψμFrequencyClasses of edgesψθ1,3γμ−1+9μE122,2γμ−1+12μE222,36γμ−1+60μ−4E333,3γμ−1+9μ−1E44


### 2.1. Computations ofK^ Banhatti Indices and Entropies for the Glycogen GLμγ Network

Generalized form of the first and secondK^ Banhatti indices is given as(2)B1λH1=∑bθψb+ψθλ=∑bw∈EH1ψb+ψθλ+ψw+ψθλ=3λ+5λγμ−1+9μ+2×4λγμ−1+12μ+5λ+6λ6γμ−1+60μ−4+2×7λγμ−1+9μ−1,B2λH1=2λ+6λγμ−1+9μ+2×4λγμ−1+12μ+6λ+9λ6γμ−1+60μ−4+2×12λγμ−1+9μ−1.The entropy of generalized firstK^ Banhatti indices for the graph of GLμγ structure is computed by using the following equation:(3)EBiλH1=log2Biλ−1BiλH1∑i=14∑biwi∈GLμγψbi+ψθiλ+ψwi+ψθiλlog2ψbi+ψθiλ+ψwi+ψθiλ.The entropy of generalized secondK^ Banhatti indices for the graph of AMμγ structure is computed by using the following equation:(4)EBiλH1=log2Biλ−1BiλH1∑i=14∑biwi∈AMμγψbi×ψθiλ+ψwi×ψθiλlog2ψbi×ψθiλ+ψwi×ψθiλ,where BiλH1 are the different K^ Banhatti indices for the graph of GLμγ. By using (1), we have the following details.


#### 2.1.1. Entropy of the FirstK^ Banhatti and Second K^ Banhatti Indices

Whenλ=1 in (2) and (3), we have(5)B11H1=3+59γ+μγ−1+4+212γ+μγ−1+5+660γ+6μγ−1−4+7+29γ+μγ−1−1=954μ+96μ−1γ−58,B21H1=2+6γμ−1+9μ+4+4γμ−1+12μ+6+96γμ−1+60μ−4+12+12γμ−1+9μ−1=1284μ+130μ−1γ−84.We are interested to compute the first and secondK^ Banhatti entropies as follows:(6)EB11H1=log2B11H1−1B11H1∑i=14ψbi+ψθi+ψwi+ψθilog2ψbi+ψθi+ψwi+ψθi=ln954μ+96μ−1γ−58ln2−504μ+48μ−1γ954μ+96μ−1γ−58+1160μ+6μ−1γ−4ln11ln2954μ+96μ−1γ−58+149μ+μ−1γ−1ln14ln2954μ+96μ−1γ−58,EB21H1=log2B21H1−1B21H1∑i=14ψbi×ψθi+ψwi×ψθilog2ψbi×ψθi+ψwi×ψθi=ln1284μ+130μ−1γ−84ln2−504μ+48μ−1γ1284μ+130μ−1γ−84+1560μ+6μ−1γ−4ln15ln21284μ+130μ−1γ−84+249μ+μ−1γ−1ln24ln21284μ+130μ−1γ−84.


#### 2.1.2. Entropy of the First and SecondK^-Hyper Banhatti Indices

Whenλ=2 in (2) and (3), we have(7)HB12H1=5232μ+530μ−1γ−342,HB22H1=10356μ+1062μ−1γ−756.Entropy of firstK^-hyper Banhatti index is given as(8)EHB12H1=log2HB12−1HB12∑i=14∑biwi∈H1ψbi2+ψθi2+ψwi2+ψθi2×log2ψbi2+ψθi2+ψwi2+ψθi2=ln5232μ+530μ−1γ−342ln2−349μ+μ−1γln34+ln25232μ+530μ−1γ−342+1920μ+160μ−1γ5232μ+530μ−1γ−342+6160μ+6μ−1γ−4ln61ln25232μ+530μ−1γ−342+989μ+μ−1γ−1ln98ln25232μ+530μ−1γ−342.Entropy of secondK^-hyper Banhatti index is given as(9)EHB22H1=log2HB22H1−1HB22H1∑i=14∑bw∈H1ψbi2×ψθi2+ψwi2×ψθi2×log2ψbi2×ψθi2+ψwi2×ψθi2=ln10356μ+1062μ−1γ−756ln2−409μ+μ−1γln40ln210356μ+1062μ−1γ−756−1920μ+160μ−1γ10356μ+1062μ−1γ−756−11760μ+6μ−1γ−4ln117ln210356μ+1062μ−1γ−756−2889μ+μ−1γ−1ln288ln210356μ+1062μ−1γ−756.


#### 2.1.3. Sum Connectivity Banhatti Entropy ofGLμγ

Whenλ=−1/2 in (2), we have(10)SB1−1/2H1=12+93+95+605+606+1807γ+1+6+13+15+65+187μγ−1+−45−46−107.Entropy of sum connectivity Banhatti index is given as(11)ESB1−1/2H1=log2SB1−1/2H1−1SB1−1/2H1∑i=14∑bw∈H11ψbi+ψθi+1ψwi+ψθi×log21ψbi+ψγi+1ψwi+ψγi=log2SB1−1/2H1−1/3+1/5γμ−1+9μln1/3+1/5ln2SB1−1/2−1/5+1/66γμ−1+60μ−4ln1/5+1/6ln2SB1−1/2+2/7γμ−1+9μ−1ln2/7ln2SB1−1/2.


#### 2.1.4. Modified First and SecondK^ Banhatti Entropies of GLμγ

Whenλ=−1 in (2), we have(12)mB1−1H1=123835μ+739210γμ−1−184105.Entropy of the modified firstK^ Banhatti index can be computed as(13)EmB1−1H1=log2mB1−1H1−1mB1−1H1∑i=14∑biwi∈H11ψbi+1ψθi+1ψwi+1ψθi×log21ψbi+1ψθi+1ψwi+1ψθi=ln1238/35μ+739/210γμ−1−184/105ln2−8γμ−1+9μln8/15151238/35μ+739/210γμ−1−184/105ln2−6μ−1/2μ−1γ1238/35μ+739/210γμ−1−184/105−116γμ−1+60μ−4ln11/30301238/35μ+739/210γμ−1−184/105ln2−29μ+μ−1γ−1ln2/771238/35μ+739/210γμ−1−184/105ln2.Whenλ=−1 in (3), we have(14)mB2−1H1=1816μ+3μ−1γ−2318.Entropy of the modified secondK^ Banhatti index can be computed as(15)EmB2−1H1=log2mB2−1H1−1mB2−1H1∑i=14∑biwi∈EH11ψbi×1ψθi+1ψwi×1ψθi×log21ψbi×1ψθi+1ψwi×1ψθi=ln181/6γ+3μγ−1−23/18ln2−29μ+μ−1γln2/33181/6γ+3μγ−1−23/18ln2+6μ+1/2μ−1γ181/6γ+3μγ−1−23/18−56γμ−1+60μ−4ln5/1818181/6γ+3μγ−1−23/18ln2+γμ−1+9μ−1ln66181/6γ+3μγ−1−23/18ln2.


#### 2.1.5. HarmonicK^ Banhatti Index of GLμγ

(16)HbH1=∑bw∈EH12ψb+ψθ+2ψw+ψθ=23+25γμ−1+9μ+24+24γμ−1+12μ+15+266γμ−1+60μ−4+27+27γμ−1+9μ−1=247635μ+739105μ−1γ−368105.Entropy of harmonicK^ Banhatti index is(17)EHbH1=log2Hb−1Hb∑i=14∑biwi∈EH12ψbi+ψθi+2ψwi+ψθilog22ψbi+ψθi+2ψwi+ψθi=log2247635μ+739105μ−1γ−368105−16γμ−1+9μln16/15152476/35γ+739/105μγ−1−368/105ln2−22/5μ−1γ+44μ−44/15ln11/152476/35γ+739/105μγ−1−368/105ln2−4/7μ−1γ+36/7μ−4/7ln4/72476/35γ+739/105μγ−1−368/105ln2.


### 2.2. Computation of Gourava Indices and Entropy Measure for the GlycogenGLμγ Network

#### 2.2.1. The First and Second Gourava Entropies ofGLμγ

The first and second Gourava indices by utilizing Table1 and Table 3 are(18)GV1H1=∑bw∈EH1ψb+ψw+ψb×ψw=1+3+1×3γμ−1+9μ+2+2+2×2γμ−1+12μ+2+3+2×36γμ−1+60μ−4+3+3+3×3γμ−1+9μ−1=96γμ−1+954μ−59,GV2H1=∑bw∈EH1ψb+ψw×ψbψw=1+3×1×3γμ−1+9μ+2+2×2×2γμ−1+12μ+2+3×2×36γμ−1+60μ−4+3+3×3×3γμ−1+9μ−1=262γμ−1+2586μ−174.Entropy of first and second Gourava indices can be computed as(19)EGV1H1=log2GV1H1−1GV1H1∑i=14∑biwi∈EH1ψbi+ψwi+ψbi×ψwi×log2ψbi+ψwi+ψbi×ψwi=ln96γμ−1+954μ−59ln2−63μ+7γμ−1ln796γμ−1+954μ−59ln2−288μ+24γμ−196γμ−1+954μ−59−660μ+66γμ−1−44ln1196γμ−1+954μ−59ln2−135μ+15γμ−1−15ln1596γμ−1+954μ−59ln2,EGV2H1=log2GV2−1GV2∑i=14∑biwi∈EH1ψbi+ψwi×ψbiψwilog2ψbi+ψwi×ψbiψwi=ln262γμ−1+2586μ−174ln2−560μ+6γμ−1−4ln5262γμ−1+2586μ−174ln2−168μ+16γμ−1262γμ−1+2586μ−174−69μ+γμ−1−1ln6262γμ−1+2586μ−174ln2−135μ+15γμ−1−15ln15262γμ−1+2586μ−174ln2.


#### 2.2.2. Product Connectivity Gourava Entropy ofGLμγ

By utilizing Table1 and Table 3, we get(20)PGVH1=∑bw∈EH11ψb+ψw×ψbψw=11+3×1×3γμ−1+9μ+12+2×2×2γμ−1+12μ+12+3×2×36γμ−1+60μ−4+13+3×3×3γμ−1+9μ−1=17.7769μ+1.77017μ−1γ−0.86636.Entropy is computed as(21)EPGVH1=log2PGVH1−1PGVH1∑i=14∑biw∈EH11ψbi+ψwi×ψbiψwi×log21ψbi+ψwi×ψbiψwi=log217.7769μ+1.77017μ−1γ−0.86636−1/12γμ−1+9μln1/1217.7769μ+1.77017μ−1γ−0.86636ln2−1/4γμ−1+12μln1/417.7769μ+1.77017μ−1γ−0.86636ln2−1/306γμ−1+60μ−4ln1/3017.7769μ+1.77017μ−1γ−0.86636ln2−1/54γμ−1+9μ−1ln1/3017.7769μ+1.77017μ−1γ−0.86636ln2.


#### 2.2.3. Sum Connectivity Gourava Entropy ofGLμγ

By utilizing Table1 and Table 3, we obtain(22)SGVH1=∑bw∈EH11ψb+ψw+ψb×ψw=11+3+1×3γμ−1+9μ+12+2+2×2γμ−1+12μ+12+3+2×36γμ−1+60μ−4+13+3+3×3γμ−1+9μ−1=28.058610μ+2.798766γμ−1−1.464334,where entropy is given as(23)ESGVH1=log2SGVH1−1SGVH1∑i=14∑biwi∈EH11ψbi+ψwi+ψbi×ψwi×log21ψbi+ψwi+ψbi×ψwi=log228.058610μ+2.798766γμ−1−1.464334−128.058610μ+2.798766γμ−1−1.464334×112γμ−1+9μlog2112+18γμ−1+12μlog218+1306γμ−1+60μ−4log2130+154γμ−1+9μ−1log2154.


#### 2.2.4. The First and Second Hyper-Gourava Entropies ofGLμγ

The first and second hyper-Gourava indices by utilizing Table1 and Table 3 are(24)GVh1H1=∑bw∈EH1ψb+ψw+ψb×ψw2=1+3+1×329γ+γμ−μ+2+2+2×2212γ+γμ−μ+2+3+2×3260γ+6γμ−6γ−4+3+3+3×329γ+γμ−μ−1=1064γμ−1−10494μ−709,GVh2H1=∑bw∈EH1ψb+ψw×ψbψw2=1+3×1×329γ+γμ−μ+2+2×2×2212γ+γμ−μ+2+3×2×3260γ+6γμ−6γ−4+3+33+3×3×329γ+γμ−μ−1=34200μ+3500γμ−1−2500,EGVh1H1=log2GVh1H1−1GVh1H1∑i=14∑biwi∈EH1ψbi+ψwi+ψbi×ψwi2×log2ψbi+ψwi+ψbi×ψwi2=ln1064γμ−1−10494μ−709ln2−11064γμ−1−10494μ−7094608μ+384γμ−1+98γμ−1+9μln7ln2+24260μ+6γμ−1−4ln11ln2+4509μ+γμ−1−1ln15ln2,EGVh2H1=log2GVh2H1−1GVh2H1∑i=14∑biwi∈EH1ψbi+ψwi×ψbiψwi2×log2ψbi+ψwi×ψbiψwi2=ln34200μ+3500γμ−1−2500ln2−134200μ+3500γμ−1−2500×ln200γμ−1+9μln10ln2+200γμ−1+12μln10ln2+2426γμ−1+60μ−4ln20ln2+1800γμ−1+9μ−1ln30ln2.


## 2.1. Computations ofK^ Banhatti Indices and Entropies for the Glycogen GLμγ Network

Generalized form of the first and secondK^ Banhatti indices is given as(2)B1λH1=∑bθψb+ψθλ=∑bw∈EH1ψb+ψθλ+ψw+ψθλ=3λ+5λγμ−1+9μ+2×4λγμ−1+12μ+5λ+6λ6γμ−1+60μ−4+2×7λγμ−1+9μ−1,B2λH1=2λ+6λγμ−1+9μ+2×4λγμ−1+12μ+6λ+9λ6γμ−1+60μ−4+2×12λγμ−1+9μ−1.The entropy of generalized firstK^ Banhatti indices for the graph of GLμγ structure is computed by using the following equation:(3)EBiλH1=log2Biλ−1BiλH1∑i=14∑biwi∈GLμγψbi+ψθiλ+ψwi+ψθiλlog2ψbi+ψθiλ+ψwi+ψθiλ.The entropy of generalized secondK^ Banhatti indices for the graph of AMμγ structure is computed by using the following equation:(4)EBiλH1=log2Biλ−1BiλH1∑i=14∑biwi∈AMμγψbi×ψθiλ+ψwi×ψθiλlog2ψbi×ψθiλ+ψwi×ψθiλ,where BiλH1 are the different K^ Banhatti indices for the graph of GLμγ. By using (1), we have the following details.


### 2.1.1. Entropy of the FirstK^ Banhatti and Second K^ Banhatti Indices

Whenλ=1 in (2) and (3), we have(5)B11H1=3+59γ+μγ−1+4+212γ+μγ−1+5+660γ+6μγ−1−4+7+29γ+μγ−1−1=954μ+96μ−1γ−58,B21H1=2+6γμ−1+9μ+4+4γμ−1+12μ+6+96γμ−1+60μ−4+12+12γμ−1+9μ−1=1284μ+130μ−1γ−84.We are interested to compute the first and secondK^ Banhatti entropies as follows:(6)EB11H1=log2B11H1−1B11H1∑i=14ψbi+ψθi+ψwi+ψθilog2ψbi+ψθi+ψwi+ψθi=ln954μ+96μ−1γ−58ln2−504μ+48μ−1γ954μ+96μ−1γ−58+1160μ+6μ−1γ−4ln11ln2954μ+96μ−1γ−58+149μ+μ−1γ−1ln14ln2954μ+96μ−1γ−58,EB21H1=log2B21H1−1B21H1∑i=14ψbi×ψθi+ψwi×ψθilog2ψbi×ψθi+ψwi×ψθi=ln1284μ+130μ−1γ−84ln2−504μ+48μ−1γ1284μ+130μ−1γ−84+1560μ+6μ−1γ−4ln15ln21284μ+130μ−1γ−84+249μ+μ−1γ−1ln24ln21284μ+130μ−1γ−84.


### 2.1.2. Entropy of the First and SecondK^-Hyper Banhatti Indices

Whenλ=2 in (2) and (3), we have(7)HB12H1=5232μ+530μ−1γ−342,HB22H1=10356μ+1062μ−1γ−756.Entropy of firstK^-hyper Banhatti index is given as(8)EHB12H1=log2HB12−1HB12∑i=14∑biwi∈H1ψbi2+ψθi2+ψwi2+ψθi2×log2ψbi2+ψθi2+ψwi2+ψθi2=ln5232μ+530μ−1γ−342ln2−349μ+μ−1γln34+ln25232μ+530μ−1γ−342+1920μ+160μ−1γ5232μ+530μ−1γ−342+6160μ+6μ−1γ−4ln61ln25232μ+530μ−1γ−342+989μ+μ−1γ−1ln98ln25232μ+530μ−1γ−342.Entropy of secondK^-hyper Banhatti index is given as(9)EHB22H1=log2HB22H1−1HB22H1∑i=14∑bw∈H1ψbi2×ψθi2+ψwi2×ψθi2×log2ψbi2×ψθi2+ψwi2×ψθi2=ln10356μ+1062μ−1γ−756ln2−409μ+μ−1γln40ln210356μ+1062μ−1γ−756−1920μ+160μ−1γ10356μ+1062μ−1γ−756−11760μ+6μ−1γ−4ln117ln210356μ+1062μ−1γ−756−2889μ+μ−1γ−1ln288ln210356μ+1062μ−1γ−756.


### 2.1.3. Sum Connectivity Banhatti Entropy ofGLμγ

Whenλ=−1/2 in (2), we have(10)SB1−1/2H1=12+93+95+605+606+1807γ+1+6+13+15+65+187μγ−1+−45−46−107.Entropy of sum connectivity Banhatti index is given as(11)ESB1−1/2H1=log2SB1−1/2H1−1SB1−1/2H1∑i=14∑bw∈H11ψbi+ψθi+1ψwi+ψθi×log21ψbi+ψγi+1ψwi+ψγi=log2SB1−1/2H1−1/3+1/5γμ−1+9μln1/3+1/5ln2SB1−1/2−1/5+1/66γμ−1+60μ−4ln1/5+1/6ln2SB1−1/2+2/7γμ−1+9μ−1ln2/7ln2SB1−1/2.


### 2.1.4. Modified First and SecondK^ Banhatti Entropies of GLμγ

Whenλ=−1 in (2), we have(12)mB1−1H1=123835μ+739210γμ−1−184105.Entropy of the modified firstK^ Banhatti index can be computed as(13)EmB1−1H1=log2mB1−1H1−1mB1−1H1∑i=14∑biwi∈H11ψbi+1ψθi+1ψwi+1ψθi×log21ψbi+1ψθi+1ψwi+1ψθi=ln1238/35μ+739/210γμ−1−184/105ln2−8γμ−1+9μln8/15151238/35μ+739/210γμ−1−184/105ln2−6μ−1/2μ−1γ1238/35μ+739/210γμ−1−184/105−116γμ−1+60μ−4ln11/30301238/35μ+739/210γμ−1−184/105ln2−29μ+μ−1γ−1ln2/771238/35μ+739/210γμ−1−184/105ln2.Whenλ=−1 in (3), we have(14)mB2−1H1=1816μ+3μ−1γ−2318.Entropy of the modified secondK^ Banhatti index can be computed as(15)EmB2−1H1=log2mB2−1H1−1mB2−1H1∑i=14∑biwi∈EH11ψbi×1ψθi+1ψwi×1ψθi×log21ψbi×1ψθi+1ψwi×1ψθi=ln181/6γ+3μγ−1−23/18ln2−29μ+μ−1γln2/33181/6γ+3μγ−1−23/18ln2+6μ+1/2μ−1γ181/6γ+3μγ−1−23/18−56γμ−1+60μ−4ln5/1818181/6γ+3μγ−1−23/18ln2+γμ−1+9μ−1ln66181/6γ+3μγ−1−23/18ln2.


### 2.1.5. HarmonicK^ Banhatti Index of GLμγ

(16)HbH1=∑bw∈EH12ψb+ψθ+2ψw+ψθ=23+25γμ−1+9μ+24+24γμ−1+12μ+15+266γμ−1+60μ−4+27+27γμ−1+9μ−1=247635μ+739105μ−1γ−368105.Entropy of harmonicK^ Banhatti index is(17)EHbH1=log2Hb−1Hb∑i=14∑biwi∈EH12ψbi+ψθi+2ψwi+ψθilog22ψbi+ψθi+2ψwi+ψθi=log2247635μ+739105μ−1γ−368105−16γμ−1+9μln16/15152476/35γ+739/105μγ−1−368/105ln2−22/5μ−1γ+44μ−44/15ln11/152476/35γ+739/105μγ−1−368/105ln2−4/7μ−1γ+36/7μ−4/7ln4/72476/35γ+739/105μγ−1−368/105ln2.


## 2.1.1. Entropy of the FirstK^ Banhatti and Second K^ Banhatti Indices

Whenλ=1 in (2) and (3), we have(5)B11H1=3+59γ+μγ−1+4+212γ+μγ−1+5+660γ+6μγ−1−4+7+29γ+μγ−1−1=954μ+96μ−1γ−58,B21H1=2+6γμ−1+9μ+4+4γμ−1+12μ+6+96γμ−1+60μ−4+12+12γμ−1+9μ−1=1284μ+130μ−1γ−84.We are interested to compute the first and secondK^ Banhatti entropies as follows:(6)EB11H1=log2B11H1−1B11H1∑i=14ψbi+ψθi+ψwi+ψθilog2ψbi+ψθi+ψwi+ψθi=ln954μ+96μ−1γ−58ln2−504μ+48μ−1γ954μ+96μ−1γ−58+1160μ+6μ−1γ−4ln11ln2954μ+96μ−1γ−58+149μ+μ−1γ−1ln14ln2954μ+96μ−1γ−58,EB21H1=log2B21H1−1B21H1∑i=14ψbi×ψθi+ψwi×ψθilog2ψbi×ψθi+ψwi×ψθi=ln1284μ+130μ−1γ−84ln2−504μ+48μ−1γ1284μ+130μ−1γ−84+1560μ+6μ−1γ−4ln15ln21284μ+130μ−1γ−84+249μ+μ−1γ−1ln24ln21284μ+130μ−1γ−84.


## 2.1.2. Entropy of the First and SecondK^-Hyper Banhatti Indices

Whenλ=2 in (2) and (3), we have(7)HB12H1=5232μ+530μ−1γ−342,HB22H1=10356μ+1062μ−1γ−756.Entropy of firstK^-hyper Banhatti index is given as(8)EHB12H1=log2HB12−1HB12∑i=14∑biwi∈H1ψbi2+ψθi2+ψwi2+ψθi2×log2ψbi2+ψθi2+ψwi2+ψθi2=ln5232μ+530μ−1γ−342ln2−349μ+μ−1γln34+ln25232μ+530μ−1γ−342+1920μ+160μ−1γ5232μ+530μ−1γ−342+6160μ+6μ−1γ−4ln61ln25232μ+530μ−1γ−342+989μ+μ−1γ−1ln98ln25232μ+530μ−1γ−342.Entropy of secondK^-hyper Banhatti index is given as(9)EHB22H1=log2HB22H1−1HB22H1∑i=14∑bw∈H1ψbi2×ψθi2+ψwi2×ψθi2×log2ψbi2×ψθi2+ψwi2×ψθi2=ln10356μ+1062μ−1γ−756ln2−409μ+μ−1γln40ln210356μ+1062μ−1γ−756−1920μ+160μ−1γ10356μ+1062μ−1γ−756−11760μ+6μ−1γ−4ln117ln210356μ+1062μ−1γ−756−2889μ+μ−1γ−1ln288ln210356μ+1062μ−1γ−756.


## 2.1.3. Sum Connectivity Banhatti Entropy ofGLμγ

Whenλ=−1/2 in (2), we have(10)SB1−1/2H1=12+93+95+605+606+1807γ+1+6+13+15+65+187μγ−1+−45−46−107.Entropy of sum connectivity Banhatti index is given as(11)ESB1−1/2H1=log2SB1−1/2H1−1SB1−1/2H1∑i=14∑bw∈H11ψbi+ψθi+1ψwi+ψθi×log21ψbi+ψγi+1ψwi+ψγi=log2SB1−1/2H1−1/3+1/5γμ−1+9μln1/3+1/5ln2SB1−1/2−1/5+1/66γμ−1+60μ−4ln1/5+1/6ln2SB1−1/2+2/7γμ−1+9μ−1ln2/7ln2SB1−1/2.


## 2.1.4. Modified First and SecondK^ Banhatti Entropies of GLμγ

Whenλ=−1 in (2), we have(12)mB1−1H1=123835μ+739210γμ−1−184105.Entropy of the modified firstK^ Banhatti index can be computed as(13)EmB1−1H1=log2mB1−1H1−1mB1−1H1∑i=14∑biwi∈H11ψbi+1ψθi+1ψwi+1ψθi×log21ψbi+1ψθi+1ψwi+1ψθi=ln1238/35μ+739/210γμ−1−184/105ln2−8γμ−1+9μln8/15151238/35μ+739/210γμ−1−184/105ln2−6μ−1/2μ−1γ1238/35μ+739/210γμ−1−184/105−116γμ−1+60μ−4ln11/30301238/35μ+739/210γμ−1−184/105ln2−29μ+μ−1γ−1ln2/771238/35μ+739/210γμ−1−184/105ln2.Whenλ=−1 in (3), we have(14)mB2−1H1=1816μ+3μ−1γ−2318.Entropy of the modified secondK^ Banhatti index can be computed as(15)EmB2−1H1=log2mB2−1H1−1mB2−1H1∑i=14∑biwi∈EH11ψbi×1ψθi+1ψwi×1ψθi×log21ψbi×1ψθi+1ψwi×1ψθi=ln181/6γ+3μγ−1−23/18ln2−29μ+μ−1γln2/33181/6γ+3μγ−1−23/18ln2+6μ+1/2μ−1γ181/6γ+3μγ−1−23/18−56γμ−1+60μ−4ln5/1818181/6γ+3μγ−1−23/18ln2+γμ−1+9μ−1ln66181/6γ+3μγ−1−23/18ln2.


## 2.1.5. HarmonicK^ Banhatti Index of GLμγ

(16)HbH1=∑bw∈EH12ψb+ψθ+2ψw+ψθ=23+25γμ−1+9μ+24+24γμ−1+12μ+15+266γμ−1+60μ−4+27+27γμ−1+9μ−1=247635μ+739105μ−1γ−368105.Entropy of harmonicK^ Banhatti index is(17)EHbH1=log2Hb−1Hb∑i=14∑biwi∈EH12ψbi+ψθi+2ψwi+ψθilog22ψbi+ψθi+2ψwi+ψθi=log2247635μ+739105μ−1γ−368105−16γμ−1+9μln16/15152476/35γ+739/105μγ−1−368/105ln2−22/5μ−1γ+44μ−44/15ln11/152476/35γ+739/105μγ−1−368/105ln2−4/7μ−1γ+36/7μ−4/7ln4/72476/35γ+739/105μγ−1−368/105ln2.


## 2.2. Computation of Gourava Indices and Entropy Measure for the GlycogenGLμγ Network

### 2.2.1. The First and Second Gourava Entropies ofGLμγ

The first and second Gourava indices by utilizing Table1 and Table 3 are(18)GV1H1=∑bw∈EH1ψb+ψw+ψb×ψw=1+3+1×3γμ−1+9μ+2+2+2×2γμ−1+12μ+2+3+2×36γμ−1+60μ−4+3+3+3×3γμ−1+9μ−1=96γμ−1+954μ−59,GV2H1=∑bw∈EH1ψb+ψw×ψbψw=1+3×1×3γμ−1+9μ+2+2×2×2γμ−1+12μ+2+3×2×36γμ−1+60μ−4+3+3×3×3γμ−1+9μ−1=262γμ−1+2586μ−174.Entropy of first and second Gourava indices can be computed as(19)EGV1H1=log2GV1H1−1GV1H1∑i=14∑biwi∈EH1ψbi+ψwi+ψbi×ψwi×log2ψbi+ψwi+ψbi×ψwi=ln96γμ−1+954μ−59ln2−63μ+7γμ−1ln796γμ−1+954μ−59ln2−288μ+24γμ−196γμ−1+954μ−59−660μ+66γμ−1−44ln1196γμ−1+954μ−59ln2−135μ+15γμ−1−15ln1596γμ−1+954μ−59ln2,EGV2H1=log2GV2−1GV2∑i=14∑biwi∈EH1ψbi+ψwi×ψbiψwilog2ψbi+ψwi×ψbiψwi=ln262γμ−1+2586μ−174ln2−560μ+6γμ−1−4ln5262γμ−1+2586μ−174ln2−168μ+16γμ−1262γμ−1+2586μ−174−69μ+γμ−1−1ln6262γμ−1+2586μ−174ln2−135μ+15γμ−1−15ln15262γμ−1+2586μ−174ln2.


### 2.2.2. Product Connectivity Gourava Entropy ofGLμγ

By utilizing Table1 and Table 3, we get(20)PGVH1=∑bw∈EH11ψb+ψw×ψbψw=11+3×1×3γμ−1+9μ+12+2×2×2γμ−1+12μ+12+3×2×36γμ−1+60μ−4+13+3×3×3γμ−1+9μ−1=17.7769μ+1.77017μ−1γ−0.86636.Entropy is computed as(21)EPGVH1=log2PGVH1−1PGVH1∑i=14∑biw∈EH11ψbi+ψwi×ψbiψwi×log21ψbi+ψwi×ψbiψwi=log217.7769μ+1.77017μ−1γ−0.86636−1/12γμ−1+9μln1/1217.7769μ+1.77017μ−1γ−0.86636ln2−1/4γμ−1+12μln1/417.7769μ+1.77017μ−1γ−0.86636ln2−1/306γμ−1+60μ−4ln1/3017.7769μ+1.77017μ−1γ−0.86636ln2−1/54γμ−1+9μ−1ln1/3017.7769μ+1.77017μ−1γ−0.86636ln2.


### 2.2.3. Sum Connectivity Gourava Entropy ofGLμγ

By utilizing Table1 and Table 3, we obtain(22)SGVH1=∑bw∈EH11ψb+ψw+ψb×ψw=11+3+1×3γμ−1+9μ+12+2+2×2γμ−1+12μ+12+3+2×36γμ−1+60μ−4+13+3+3×3γμ−1+9μ−1=28.058610μ+2.798766γμ−1−1.464334,where entropy is given as(23)ESGVH1=log2SGVH1−1SGVH1∑i=14∑biwi∈EH11ψbi+ψwi+ψbi×ψwi×log21ψbi+ψwi+ψbi×ψwi=log228.058610μ+2.798766γμ−1−1.464334−128.058610μ+2.798766γμ−1−1.464334×112γμ−1+9μlog2112+18γμ−1+12μlog218+1306γμ−1+60μ−4log2130+154γμ−1+9μ−1log2154.


### 2.2.4. The First and Second Hyper-Gourava Entropies ofGLμγ

The first and second hyper-Gourava indices by utilizing Table1 and Table 3 are(24)GVh1H1=∑bw∈EH1ψb+ψw+ψb×ψw2=1+3+1×329γ+γμ−μ+2+2+2×2212γ+γμ−μ+2+3+2×3260γ+6γμ−6γ−4+3+3+3×329γ+γμ−μ−1=1064γμ−1−10494μ−709,GVh2H1=∑bw∈EH1ψb+ψw×ψbψw2=1+3×1×329γ+γμ−μ+2+2×2×2212γ+γμ−μ+2+3×2×3260γ+6γμ−6γ−4+3+33+3×3×329γ+γμ−μ−1=34200μ+3500γμ−1−2500,EGVh1H1=log2GVh1H1−1GVh1H1∑i=14∑biwi∈EH1ψbi+ψwi+ψbi×ψwi2×log2ψbi+ψwi+ψbi×ψwi2=ln1064γμ−1−10494μ−709ln2−11064γμ−1−10494μ−7094608μ+384γμ−1+98γμ−1+9μln7ln2+24260μ+6γμ−1−4ln11ln2+4509μ+γμ−1−1ln15ln2,EGVh2H1=log2GVh2H1−1GVh2H1∑i=14∑biwi∈EH1ψbi+ψwi×ψbiψwi2×log2ψbi+ψwi×ψbiψwi2=ln34200μ+3500γμ−1−2500ln2−134200μ+3500γμ−1−2500×ln200γμ−1+9μln10ln2+200γμ−1+12μln10ln2+2426γμ−1+60μ−4ln20ln2+1800γμ−1+9μ−1ln30ln2.


## 2.2.1. The First and Second Gourava Entropies ofGLμγ

The first and second Gourava indices by utilizing Table1 and Table 3 are(18)GV1H1=∑bw∈EH1ψb+ψw+ψb×ψw=1+3+1×3γμ−1+9μ+2+2+2×2γμ−1+12μ+2+3+2×36γμ−1+60μ−4+3+3+3×3γμ−1+9μ−1=96γμ−1+954μ−59,GV2H1=∑bw∈EH1ψb+ψw×ψbψw=1+3×1×3γμ−1+9μ+2+2×2×2γμ−1+12μ+2+3×2×36γμ−1+60μ−4+3+3×3×3γμ−1+9μ−1=262γμ−1+2586μ−174.Entropy of first and second Gourava indices can be computed as(19)EGV1H1=log2GV1H1−1GV1H1∑i=14∑biwi∈EH1ψbi+ψwi+ψbi×ψwi×log2ψbi+ψwi+ψbi×ψwi=ln96γμ−1+954μ−59ln2−63μ+7γμ−1ln796γμ−1+954μ−59ln2−288μ+24γμ−196γμ−1+954μ−59−660μ+66γμ−1−44ln1196γμ−1+954μ−59ln2−135μ+15γμ−1−15ln1596γμ−1+954μ−59ln2,EGV2H1=log2GV2−1GV2∑i=14∑biwi∈EH1ψbi+ψwi×ψbiψwilog2ψbi+ψwi×ψbiψwi=ln262γμ−1+2586μ−174ln2−560μ+6γμ−1−4ln5262γμ−1+2586μ−174ln2−168μ+16γμ−1262γμ−1+2586μ−174−69μ+γμ−1−1ln6262γμ−1+2586μ−174ln2−135μ+15γμ−1−15ln15262γμ−1+2586μ−174ln2.


## 2.2.2. Product Connectivity Gourava Entropy ofGLμγ

By utilizing Table1 and Table 3, we get(20)PGVH1=∑bw∈EH11ψb+ψw×ψbψw=11+3×1×3γμ−1+9μ+12+2×2×2γμ−1+12μ+12+3×2×36γμ−1+60μ−4+13+3×3×3γμ−1+9μ−1=17.7769μ+1.77017μ−1γ−0.86636.Entropy is computed as(21)EPGVH1=log2PGVH1−1PGVH1∑i=14∑biw∈EH11ψbi+ψwi×ψbiψwi×log21ψbi+ψwi×ψbiψwi=log217.7769μ+1.77017μ−1γ−0.86636−1/12γμ−1+9μln1/1217.7769μ+1.77017μ−1γ−0.86636ln2−1/4γμ−1+12μln1/417.7769μ+1.77017μ−1γ−0.86636ln2−1/306γμ−1+60μ−4ln1/3017.7769μ+1.77017μ−1γ−0.86636ln2−1/54γμ−1+9μ−1ln1/3017.7769μ+1.77017μ−1γ−0.86636ln2.


## 2.2.3. Sum Connectivity Gourava Entropy ofGLμγ

By utilizing Table1 and Table 3, we obtain(22)SGVH1=∑bw∈EH11ψb+ψw+ψb×ψw=11+3+1×3γμ−1+9μ+12+2+2×2γμ−1+12μ+12+3+2×36γμ−1+60μ−4+13+3+3×3γμ−1+9μ−1=28.058610μ+2.798766γμ−1−1.464334,where entropy is given as(23)ESGVH1=log2SGVH1−1SGVH1∑i=14∑biwi∈EH11ψbi+ψwi+ψbi×ψwi×log21ψbi+ψwi+ψbi×ψwi=log228.058610μ+2.798766γμ−1−1.464334−128.058610μ+2.798766γμ−1−1.464334×112γμ−1+9μlog2112+18γμ−1+12μlog218+1306γμ−1+60μ−4log2130+154γμ−1+9μ−1log2154.


## 2.2.4. The First and Second Hyper-Gourava Entropies ofGLμγ

The first and second hyper-Gourava indices by utilizing Table1 and Table 3 are(24)GVh1H1=∑bw∈EH1ψb+ψw+ψb×ψw2=1+3+1×329γ+γμ−μ+2+2+2×2212γ+γμ−μ+2+3+2×3260γ+6γμ−6γ−4+3+3+3×329γ+γμ−μ−1=1064γμ−1−10494μ−709,GVh2H1=∑bw∈EH1ψb+ψw×ψbψw2=1+3×1×329γ+γμ−μ+2+2×2×2212γ+γμ−μ+2+3×2×3260γ+6γμ−6γ−4+3+33+3×3×329γ+γμ−μ−1=34200μ+3500γμ−1−2500,EGVh1H1=log2GVh1H1−1GVh1H1∑i=14∑biwi∈EH1ψbi+ψwi+ψbi×ψwi2×log2ψbi+ψwi+ψbi×ψwi2=ln1064γμ−1−10494μ−709ln2−11064γμ−1−10494μ−7094608μ+384γμ−1+98γμ−1+9μln7ln2+24260μ+6γμ−1−4ln11ln2+4509μ+γμ−1−1ln15ln2,EGVh2H1=log2GVh2H1−1GVh2H1∑i=14∑biwi∈EH1ψbi+ψwi×ψbiψwi2×log2ψbi+ψwi×ψbiψwi2=ln34200μ+3500γμ−1−2500ln2−134200μ+3500γμ−1−2500×ln200γμ−1+9μln10ln2+200γμ−1+12μln10ln2+2426γμ−1+60μ−4ln20ln2+1800γμ−1+9μ−1ln30ln2.


## 3. Formation of the AmylopectinAMμγ Network

In comparison to glycogen, amylopectin has fewer branches and is less compact. These molecules have an open structure due to the helical branching structure; as a result, enzymes may easily access them. As a result, they can be quickly disassembled or reassembled. The branch in amylopectin is formed viaαC1−C6 glycosidic bonding, which happens around every 20 glucose residues. The primary distinction between amylopectin and glycogen is that amylopectin is insoluble, whereas glycogen is soluble. Amylopectin is one of two kinds of starch that make up the majority of plant storage polysaccharides. In mammals, glycogen is the primary storage polysaccharide. The linear chain of both amylopectin and glycogen is made up of αC1−C4 glycosidic links, whereas the branches are made up of αC1−C6 glycosidic linkages. Amylopectin is a protein that is utilized in the textile, paper, and laundry industries. It is also utilized in the food business and in the making of paper boards and adhesives for paper in breweries as a raw material. It is used in soaps and a variety of desserts as a thickening and gelling ingredient. The amylopectin molecule is used to build the AMμγ network, which has μ−1 branches with lengths γ, where 2≤γ≤19. After some basic computation, we can see that ∣VAMμγ∣=8γμ−1+160μ−4 and ∣EAMμγ∣=9γμ−1+180μ−5. The principle strategy utilized here is the way to deal with edge partitioning and vertices degree calculating, as given in Table 4.Table 4
Edge description ofAMμγ.
ψb,ψwFrequencyClasses of edgesψθ1,3γμ−1+19μE122,2γμ−1+22μE222,36γμ−1+120μ−4E333,3γμ−1+19μ−1E44


### 3.1.K^ Banhatti Indices and Entropy Measures for the AMμγ Network

Generalized form of the firstK^ Banhatti index by using Table 4 is given as(25)BιλH2=∑bwψb+ψwλ=3λ+5λγμ−1+19μ+4λ+2λγμ−1+22μ+5λ+6λ6γμ−1+120μ−4+7λ+2λγμ−1+19μ−1.Generalized form of the secondK^ Banhatti index with the help of Table 4 is given as(26)BιλH2=∑bwψb+ψwλ=2λ+6λγμ−1+19μ+4λ+2λγμ−1+22μ+6λ+9λ6γμ−1+120μ−4+12λ+2λγμ−1+19μ−1.The entropy of generalized firstK^ Banhatti indices for the graph of AMμγ structure is computed by using the following equation:(27)EBiλH2=log2Biλ−1BiλH2∑i=14∑biwi∈AMμγψbi+ψθiλ+ψwi+ψθiλlog2ψbi+ψθiλ+ψwi+ψθiλ.The entropy of generalized secondK^ Banhatti indices for the graph of AMμγ structure is computed by using the following equation:(28)EBiλH2=log2Biλ−1BiλH2∑i=14∑biwi∈AMμγψbi×ψθiλ+ψwi×ψθiλlog2ψbi×ψθiλ+ψwi×ψθiλ,where BiλH2 are the different K^ Banhatti indices for the graph of AMμγ.


#### 3.1.1. The First and SecondK^ Banhatti Entropies of AMμγ

In both (25) and (26), for λ=1, we have(29)B11H2=1914μ+96μ−1γ−58,B21H2=25584μ+1062μ−1γ−84.Corresponding entropies by using (27) and (28) for λ=1 are(30)EB21H2=ln1914μ+96μ−1γ−58ln2−984μ+48μ−1γ1914μ+96μ−1γ−58+11120μ+6μ−1γ−4ln11n21914μ+96μ−1γ−58+1419μ+μ−1γ−1ln14ln21914μ+96μ−1γ−58,EB21H2=ln2584μ+130μ−1γ−84ln2−984μ+48μ−1γ2584μ+130μ−1γ−84+15120μ+6μ−1γ−4ln15ln22584μ+130μ−1γ−84+2419μ+μ−1γ−1ln24ln22584μ+130μ−1γ−84.


#### 3.1.2. Entropy of the First and SecondK^-Hyper Banhatti Indices

In both (25) and (26), for λ=2, we have(31)HB12H2=10532μ+530μ−1γ−342,HB22H2=20976μ+1062μ−1γ−756.Entropy of the first and secondK^-hyper Banhatti indices by using both (27) and (28) for λ=2 is given as(32)EHB12H2=ln10532μ+530μ−1γ−342ln2−3419μ+μ−1γln34+ln210532μ+530μ−1γ−342+3520μ+160μ−1γ10532μ+530μ−1γ−342+61120μ+6μ−1γ−4ln61ln210532μ+530μ−1γ−342+9819μ+μ−1γ−1ln98ln210532μ+530μ−1γ−342,EHB22H2=ln20976μ+1062μ−1γ−756ln2−4019μ+μ−1γln40ln220976μ+1062μ−1γ−756−3520μ+160μ−1γ20976μ+1062μ−1γ−756−117120μ+6μ−1γ−4ln117ln220976μ+1062μ−1γ−756−28819μ+μ−1γ−1ln288ln220976μ+1062μ−1γ−756.


#### 3.1.3. Entropy of Sum Connectivity Banhatti Index forAMμγ

Whenλ=−1/2 in equation (6), we have(33)SB1−1/2H2=13+15γμ−1+19μ+1γμ−1+22μ+15+166γμ−1+120μ−4+27γμ−1+19μ−1=158.4847717μ+7.91326318μ−1γ−4.17777642.Entropy of sum connectivity Banhatti index by using (27) is given as(34)ESB1−1/2H2=log2SB1−1/2H1−1/3+1/5γμ−1+19μln1/3+1/5ln2SB1−1/2−1/5+1/66γμ−1+120μ−4ln1/5+1/6ln2SB1−1/2−2/7γμ−1+19μ−1ln2/7ln2SB1−1/2.


#### 3.1.4. Modified First and SecondK^ Banhatti Entropies of AMμγ

In both (25) and (26), for λ=−1, we have(35)mB1−1H2=13+15γμ−1+19μ+1γμ−1+22μ+15+166γμ−1+120μ−4+27γμ−1+19μ−1=7409105μ+739210γμ−1−184105,mB1−1H2=12+16γμ−1+19μ+14+14γμ−1+22μ+16+196γμ−1+120μ−4+112+112γμ−1+19μ−1=3616μ+3γμ−1−2318.Entropy of modified first and secondK^ Banhatti indices by using (27) and (28) can be computed as(36)EmB1−1H2=ln7409/105μ+739/210γμ−1−184/105ln2−8γμ−1+19μln8/15157409/105μ+739/210γμ−1−184/105ln2−11μ−1/2μ−1γ7409/105μ+739/210γμ−1−184/105−116γμ−1+120μ−4ln11/30307409/105μ+739/210γμ−1−184/105ln2−219μ+μ−1γ−1ln2/777409/105μ+739/210γμ−1−184/105ln2,EmB2−1H2=ln361/6μ+3γμ−1−23/18ln2−219μ+μ−1γln2/33361/6μ+3γμ−1−23/18ln2+11μ+1/2μ−1γ361/6μ+3γμ−1−23/18−56γμ−1+120μ−4ln5/1818361/6μ+3γμ−1−23/18ln2+γμ−1+19μ−1ln66361/6μ+3γμ−1−23/18ln2.


#### 3.1.5. HarmonicK^ Banhatti Index of AMμγ

(37)HbH2=∑bw∈EH22ψb+ψθ+2ψw+ψθ=23+2519γ+μγ−1+24+2422γ+μγ−1+15+26120γ+6μγ−1−4+27+2719γ+μγ−1−1=1488105γ+739105μγ−1−368105.


### 3.2. Gourava Indices and Entropy Measures for the AmylopectinAMρλ Network

#### 3.2.1. First and Second Gourava Entropies ofAMμγ

First and second Gourava indices are calculated by using Table4:(38)GO1H2=∑bw∈EH2ψb+ψw+ψb×ψw=1+3+1×3γμ−1+19μ+2+2+2×2γμ−1+22μ+2+3+2×36γμ−1+120μ−4+3+3+3×3γμ−1+19μ−1=1914μ+96γμ−1−59,GO2H2=∑bw∈EH2ψb+ψw×ψbψw=1+3×1×3γμ−1+19μ+2+2×2×2γμ−1+22μ+2+3×2×36γμ−1+120μ−4+3+3×3×3γμ−1+19μ−1=8716γμ−1+84612μ−6516.Entropy of first and second Gourava indices by using Table4 and (1) are(39)EGO1H2=ln1914μ+96γμ−1−59ln2−7γμ−1+19μln71914μ+96γμ−1−59ln2−8γμ−1+22μln81914μ+96γμ−1−59ln2−116γμ−1+120μ−4ln111914μ+96γμ−1−59ln2−15γμ−1+19μ−1ln151914μ+96γμ−1−59ln2,EGO2H2=ln8716γμ−1+84612μ−6516ln2−12γμ−1+19μln128716γμ−1+84612μ−6516ln2−16γμ−1+22μln168716γμ−1+84612μ−6516ln2−306γμ−1+120μ−4ln308716γμ−1+84612μ−6516ln2−54γμ−1+19μ−1ln548716γμ−1+84612μ−6516ln2.


#### 3.2.2. Product Connectivity Gourava Entropy ofAMμγ

Product connectivity Gourava index and entropy with the help of Table4 and (1) can be calculated as(40)PGVH1=11+3×1×3γμ−1+19μ+12+2×2×2γμ−1+22μ+12+3×2×36γμ−1+120μ−4+13+3×3×3γμ−1+19μ−1=35.47865μ+1.77017γμ−1−0.86636,EPGVH1=log235.47865γ+1.77017μγ−1−0.86636−135.47865γ+1.77017μγ−1−0.86636×112γμ−1+19μlog2112+14γμ−1+22μlog214+1306γμ−1+120μ−4log2130+154γμ−1+19μ−1log2154.


#### 3.2.3. Sum Connectivity Gourava Index and Entropy ofAMμγ

(41)SGVH2=∑γμ∈EH11ψγ+ψμ+ψγ×ψμ=11+3+1×3γμ−1+19μ+12+2+2×2γμ−1+22μ+12+3+2×36γμ−1+120μ−4+13+3+3×3γμ−1+19μ−1=28.058610μ+2.798766γμ−1−1.464334,ESGVH2=log228.058610γ+2.798766μγ−1−1.464334−128.058610γ+2.798766μγ−1−1.464334×17γμ−1+19μlog217+18γμ−1+22μlog218+1116γμ−1+120μ−4log2111+115γμ−1+19μ−1log2115.


#### 3.2.4. First and Second Hyper-Gourava Entropies ofAMμγ

First and second hyper-Gourava indices with the help of Table4 are(42)HGO1H2=∑bw∈EH2ψb+ψw+ψb×ψw2=1+3+1×32γμ−1+19μ+2+2+2×22γμ−1+22μ+2+3+2×326γμ−1+120μ−4+3+3+3×32γμ−1+19μ−1=10494μ+1064γμ−1−709,HGO2H2=∑bw∈EH2ψb+ψw×ψbψw2=1+3×1×32γμ−1+19μ+2+2×2×22γμ−1+22μ+2+3×2×326γμ−1+120μ−4+3+3×3×32γμ−1+19μ−1=4340μ+218γμ−1−136.Entropy of first and second hyper-Gourava Indices is computed as(43)EHGO1H2=ln10494μ+1064γμ−1−709ln2−110494μ+1064γμ−1−709×4608μ+384γμ−1+989μ+γμ−1ln7ln2+24260μ+6γμ−1−4ln11ln2+4509μ+γμ−1−1ln15ln2,EHGO2H2=ln4340μ+218γμ−1−136ln2−14340μ+218γμ−1−136×2624μ+128γμ−1+50120μ+6γμ−1−4ln5ln2+7219μ+γμ−1−1ln6ln2.


## 3.1.K^ Banhatti Indices and Entropy Measures for the AMμγ Network

Generalized form of the firstK^ Banhatti index by using Table 4 is given as(25)BιλH2=∑bwψb+ψwλ=3λ+5λγμ−1+19μ+4λ+2λγμ−1+22μ+5λ+6λ6γμ−1+120μ−4+7λ+2λγμ−1+19μ−1.Generalized form of the secondK^ Banhatti index with the help of Table 4 is given as(26)BιλH2=∑bwψb+ψwλ=2λ+6λγμ−1+19μ+4λ+2λγμ−1+22μ+6λ+9λ6γμ−1+120μ−4+12λ+2λγμ−1+19μ−1.The entropy of generalized firstK^ Banhatti indices for the graph of AMμγ structure is computed by using the following equation:(27)EBiλH2=log2Biλ−1BiλH2∑i=14∑biwi∈AMμγψbi+ψθiλ+ψwi+ψθiλlog2ψbi+ψθiλ+ψwi+ψθiλ.The entropy of generalized secondK^ Banhatti indices for the graph of AMμγ structure is computed by using the following equation:(28)EBiλH2=log2Biλ−1BiλH2∑i=14∑biwi∈AMμγψbi×ψθiλ+ψwi×ψθiλlog2ψbi×ψθiλ+ψwi×ψθiλ,where BiλH2 are the different K^ Banhatti indices for the graph of AMμγ.


### 3.1.1. The First and SecondK^ Banhatti Entropies of AMμγ

In both (25) and (26), for λ=1, we have(29)B11H2=1914μ+96μ−1γ−58,B21H2=25584μ+1062μ−1γ−84.Corresponding entropies by using (27) and (28) for λ=1 are(30)EB21H2=ln1914μ+96μ−1γ−58ln2−984μ+48μ−1γ1914μ+96μ−1γ−58+11120μ+6μ−1γ−4ln11n21914μ+96μ−1γ−58+1419μ+μ−1γ−1ln14ln21914μ+96μ−1γ−58,EB21H2=ln2584μ+130μ−1γ−84ln2−984μ+48μ−1γ2584μ+130μ−1γ−84+15120μ+6μ−1γ−4ln15ln22584μ+130μ−1γ−84+2419μ+μ−1γ−1ln24ln22584μ+130μ−1γ−84.


### 3.1.2. Entropy of the First and SecondK^-Hyper Banhatti Indices

In both (25) and (26), for λ=2, we have(31)HB12H2=10532μ+530μ−1γ−342,HB22H2=20976μ+1062μ−1γ−756.Entropy of the first and secondK^-hyper Banhatti indices by using both (27) and (28) for λ=2 is given as(32)EHB12H2=ln10532μ+530μ−1γ−342ln2−3419μ+μ−1γln34+ln210532μ+530μ−1γ−342+3520μ+160μ−1γ10532μ+530μ−1γ−342+61120μ+6μ−1γ−4ln61ln210532μ+530μ−1γ−342+9819μ+μ−1γ−1ln98ln210532μ+530μ−1γ−342,EHB22H2=ln20976μ+1062μ−1γ−756ln2−4019μ+μ−1γln40ln220976μ+1062μ−1γ−756−3520μ+160μ−1γ20976μ+1062μ−1γ−756−117120μ+6μ−1γ−4ln117ln220976μ+1062μ−1γ−756−28819μ+μ−1γ−1ln288ln220976μ+1062μ−1γ−756.


### 3.1.3. Entropy of Sum Connectivity Banhatti Index forAMμγ

Whenλ=−1/2 in equation (6), we have(33)SB1−1/2H2=13+15γμ−1+19μ+1γμ−1+22μ+15+166γμ−1+120μ−4+27γμ−1+19μ−1=158.4847717μ+7.91326318μ−1γ−4.17777642.Entropy of sum connectivity Banhatti index by using (27) is given as(34)ESB1−1/2H2=log2SB1−1/2H1−1/3+1/5γμ−1+19μln1/3+1/5ln2SB1−1/2−1/5+1/66γμ−1+120μ−4ln1/5+1/6ln2SB1−1/2−2/7γμ−1+19μ−1ln2/7ln2SB1−1/2.


### 3.1.4. Modified First and SecondK^ Banhatti Entropies of AMμγ

In both (25) and (26), for λ=−1, we have(35)mB1−1H2=13+15γμ−1+19μ+1γμ−1+22μ+15+166γμ−1+120μ−4+27γμ−1+19μ−1=7409105μ+739210γμ−1−184105,mB1−1H2=12+16γμ−1+19μ+14+14γμ−1+22μ+16+196γμ−1+120μ−4+112+112γμ−1+19μ−1=3616μ+3γμ−1−2318.Entropy of modified first and secondK^ Banhatti indices by using (27) and (28) can be computed as(36)EmB1−1H2=ln7409/105μ+739/210γμ−1−184/105ln2−8γμ−1+19μln8/15157409/105μ+739/210γμ−1−184/105ln2−11μ−1/2μ−1γ7409/105μ+739/210γμ−1−184/105−116γμ−1+120μ−4ln11/30307409/105μ+739/210γμ−1−184/105ln2−219μ+μ−1γ−1ln2/777409/105μ+739/210γμ−1−184/105ln2,EmB2−1H2=ln361/6μ+3γμ−1−23/18ln2−219μ+μ−1γln2/33361/6μ+3γμ−1−23/18ln2+11μ+1/2μ−1γ361/6μ+3γμ−1−23/18−56γμ−1+120μ−4ln5/1818361/6μ+3γμ−1−23/18ln2+γμ−1+19μ−1ln66361/6μ+3γμ−1−23/18ln2.


### 3.1.5. HarmonicK^ Banhatti Index of AMμγ

(37)HbH2=∑bw∈EH22ψb+ψθ+2ψw+ψθ=23+2519γ+μγ−1+24+2422γ+μγ−1+15+26120γ+6μγ−1−4+27+2719γ+μγ−1−1=1488105γ+739105μγ−1−368105.


## 3.1.1. The First and SecondK^ Banhatti Entropies of AMμγ

In both (25) and (26), for λ=1, we have(29)B11H2=1914μ+96μ−1γ−58,B21H2=25584μ+1062μ−1γ−84.Corresponding entropies by using (27) and (28) for λ=1 are(30)EB21H2=ln1914μ+96μ−1γ−58ln2−984μ+48μ−1γ1914μ+96μ−1γ−58+11120μ+6μ−1γ−4ln11n21914μ+96μ−1γ−58+1419μ+μ−1γ−1ln14ln21914μ+96μ−1γ−58,EB21H2=ln2584μ+130μ−1γ−84ln2−984μ+48μ−1γ2584μ+130μ−1γ−84+15120μ+6μ−1γ−4ln15ln22584μ+130μ−1γ−84+2419μ+μ−1γ−1ln24ln22584μ+130μ−1γ−84.


## 3.1.2. Entropy of the First and SecondK^-Hyper Banhatti Indices

In both (25) and (26), for λ=2, we have(31)HB12H2=10532μ+530μ−1γ−342,HB22H2=20976μ+1062μ−1γ−756.Entropy of the first and secondK^-hyper Banhatti indices by using both (27) and (28) for λ=2 is given as(32)EHB12H2=ln10532μ+530μ−1γ−342ln2−3419μ+μ−1γln34+ln210532μ+530μ−1γ−342+3520μ+160μ−1γ10532μ+530μ−1γ−342+61120μ+6μ−1γ−4ln61ln210532μ+530μ−1γ−342+9819μ+μ−1γ−1ln98ln210532μ+530μ−1γ−342,EHB22H2=ln20976μ+1062μ−1γ−756ln2−4019μ+μ−1γln40ln220976μ+1062μ−1γ−756−3520μ+160μ−1γ20976μ+1062μ−1γ−756−117120μ+6μ−1γ−4ln117ln220976μ+1062μ−1γ−756−28819μ+μ−1γ−1ln288ln220976μ+1062μ−1γ−756.


## 3.1.3. Entropy of Sum Connectivity Banhatti Index forAMμγ

Whenλ=−1/2 in equation (6), we have(33)SB1−1/2H2=13+15γμ−1+19μ+1γμ−1+22μ+15+166γμ−1+120μ−4+27γμ−1+19μ−1=158.4847717μ+7.91326318μ−1γ−4.17777642.Entropy of sum connectivity Banhatti index by using (27) is given as(34)ESB1−1/2H2=log2SB1−1/2H1−1/3+1/5γμ−1+19μln1/3+1/5ln2SB1−1/2−1/5+1/66γμ−1+120μ−4ln1/5+1/6ln2SB1−1/2−2/7γμ−1+19μ−1ln2/7ln2SB1−1/2.


## 3.1.4. Modified First and SecondK^ Banhatti Entropies of AMμγ

In both (25) and (26), for λ=−1, we have(35)mB1−1H2=13+15γμ−1+19μ+1γμ−1+22μ+15+166γμ−1+120μ−4+27γμ−1+19μ−1=7409105μ+739210γμ−1−184105,mB1−1H2=12+16γμ−1+19μ+14+14γμ−1+22μ+16+196γμ−1+120μ−4+112+112γμ−1+19μ−1=3616μ+3γμ−1−2318.Entropy of modified first and secondK^ Banhatti indices by using (27) and (28) can be computed as(36)EmB1−1H2=ln7409/105μ+739/210γμ−1−184/105ln2−8γμ−1+19μln8/15157409/105μ+739/210γμ−1−184/105ln2−11μ−1/2μ−1γ7409/105μ+739/210γμ−1−184/105−116γμ−1+120μ−4ln11/30307409/105μ+739/210γμ−1−184/105ln2−219μ+μ−1γ−1ln2/777409/105μ+739/210γμ−1−184/105ln2,EmB2−1H2=ln361/6μ+3γμ−1−23/18ln2−219μ+μ−1γln2/33361/6μ+3γμ−1−23/18ln2+11μ+1/2μ−1γ361/6μ+3γμ−1−23/18−56γμ−1+120μ−4ln5/1818361/6μ+3γμ−1−23/18ln2+γμ−1+19μ−1ln66361/6μ+3γμ−1−23/18ln2.


## 3.1.5. HarmonicK^ Banhatti Index of AMμγ

(37)HbH2=∑bw∈EH22ψb+ψθ+2ψw+ψθ=23+2519γ+μγ−1+24+2422γ+μγ−1+15+26120γ+6μγ−1−4+27+2719γ+μγ−1−1=1488105γ+739105μγ−1−368105.


## 3.2. Gourava Indices and Entropy Measures for the AmylopectinAMρλ Network

### 3.2.1. First and Second Gourava Entropies ofAMμγ

First and second Gourava indices are calculated by using Table4:(38)GO1H2=∑bw∈EH2ψb+ψw+ψb×ψw=1+3+1×3γμ−1+19μ+2+2+2×2γμ−1+22μ+2+3+2×36γμ−1+120μ−4+3+3+3×3γμ−1+19μ−1=1914μ+96γμ−1−59,GO2H2=∑bw∈EH2ψb+ψw×ψbψw=1+3×1×3γμ−1+19μ+2+2×2×2γμ−1+22μ+2+3×2×36γμ−1+120μ−4+3+3×3×3γμ−1+19μ−1=8716γμ−1+84612μ−6516.Entropy of first and second Gourava indices by using Table4 and (1) are(39)EGO1H2=ln1914μ+96γμ−1−59ln2−7γμ−1+19μln71914μ+96γμ−1−59ln2−8γμ−1+22μln81914μ+96γμ−1−59ln2−116γμ−1+120μ−4ln111914μ+96γμ−1−59ln2−15γμ−1+19μ−1ln151914μ+96γμ−1−59ln2,EGO2H2=ln8716γμ−1+84612μ−6516ln2−12γμ−1+19μln128716γμ−1+84612μ−6516ln2−16γμ−1+22μln168716γμ−1+84612μ−6516ln2−306γμ−1+120μ−4ln308716γμ−1+84612μ−6516ln2−54γμ−1+19μ−1ln548716γμ−1+84612μ−6516ln2.


### 3.2.2. Product Connectivity Gourava Entropy ofAMμγ

Product connectivity Gourava index and entropy with the help of Table4 and (1) can be calculated as(40)PGVH1=11+3×1×3γμ−1+19μ+12+2×2×2γμ−1+22μ+12+3×2×36γμ−1+120μ−4+13+3×3×3γμ−1+19μ−1=35.47865μ+1.77017γμ−1−0.86636,EPGVH1=log235.47865γ+1.77017μγ−1−0.86636−135.47865γ+1.77017μγ−1−0.86636×112γμ−1+19μlog2112+14γμ−1+22μlog214+1306γμ−1+120μ−4log2130+154γμ−1+19μ−1log2154.


### 3.2.3. Sum Connectivity Gourava Index and Entropy ofAMμγ

(41)SGVH2=∑γμ∈EH11ψγ+ψμ+ψγ×ψμ=11+3+1×3γμ−1+19μ+12+2+2×2γμ−1+22μ+12+3+2×36γμ−1+120μ−4+13+3+3×3γμ−1+19μ−1=28.058610μ+2.798766γμ−1−1.464334,ESGVH2=log228.058610γ+2.798766μγ−1−1.464334−128.058610γ+2.798766μγ−1−1.464334×17γμ−1+19μlog217+18γμ−1+22μlog218+1116γμ−1+120μ−4log2111+115γμ−1+19μ−1log2115.


### 3.2.4. First and Second Hyper-Gourava Entropies ofAMμγ

First and second hyper-Gourava indices with the help of Table4 are(42)HGO1H2=∑bw∈EH2ψb+ψw+ψb×ψw2=1+3+1×32γμ−1+19μ+2+2+2×22γμ−1+22μ+2+3+2×326γμ−1+120μ−4+3+3+3×32γμ−1+19μ−1=10494μ+1064γμ−1−709,HGO2H2=∑bw∈EH2ψb+ψw×ψbψw2=1+3×1×32γμ−1+19μ+2+2×2×22γμ−1+22μ+2+3×2×326γμ−1+120μ−4+3+3×3×32γμ−1+19μ−1=4340μ+218γμ−1−136.Entropy of first and second hyper-Gourava Indices is computed as(43)EHGO1H2=ln10494μ+1064γμ−1−709ln2−110494μ+1064γμ−1−709×4608μ+384γμ−1+989μ+γμ−1ln7ln2+24260μ+6γμ−1−4ln11ln2+4509μ+γμ−1−1ln15ln2,EHGO2H2=ln4340μ+218γμ−1−136ln2−14340μ+218γμ−1−136×2624μ+128γμ−1+50120μ+6γμ−1−4ln5ln2+7219μ+γμ−1−1ln6ln2.


## 3.2.1. First and Second Gourava Entropies ofAMμγ

First and second Gourava indices are calculated by using Table4:(38)GO1H2=∑bw∈EH2ψb+ψw+ψb×ψw=1+3+1×3γμ−1+19μ+2+2+2×2γμ−1+22μ+2+3+2×36γμ−1+120μ−4+3+3+3×3γμ−1+19μ−1=1914μ+96γμ−1−59,GO2H2=∑bw∈EH2ψb+ψw×ψbψw=1+3×1×3γμ−1+19μ+2+2×2×2γμ−1+22μ+2+3×2×36γμ−1+120μ−4+3+3×3×3γμ−1+19μ−1=8716γμ−1+84612μ−6516.Entropy of first and second Gourava indices by using Table4 and (1) are(39)EGO1H2=ln1914μ+96γμ−1−59ln2−7γμ−1+19μln71914μ+96γμ−1−59ln2−8γμ−1+22μln81914μ+96γμ−1−59ln2−116γμ−1+120μ−4ln111914μ+96γμ−1−59ln2−15γμ−1+19μ−1ln151914μ+96γμ−1−59ln2,EGO2H2=ln8716γμ−1+84612μ−6516ln2−12γμ−1+19μln128716γμ−1+84612μ−6516ln2−16γμ−1+22μln168716γμ−1+84612μ−6516ln2−306γμ−1+120μ−4ln308716γμ−1+84612μ−6516ln2−54γμ−1+19μ−1ln548716γμ−1+84612μ−6516ln2.


## 3.2.2. Product Connectivity Gourava Entropy ofAMμγ

Product connectivity Gourava index and entropy with the help of Table4 and (1) can be calculated as(40)PGVH1=11+3×1×3γμ−1+19μ+12+2×2×2γμ−1+22μ+12+3×2×36γμ−1+120μ−4+13+3×3×3γμ−1+19μ−1=35.47865μ+1.77017γμ−1−0.86636,EPGVH1=log235.47865γ+1.77017μγ−1−0.86636−135.47865γ+1.77017μγ−1−0.86636×112γμ−1+19μlog2112+14γμ−1+22μlog214+1306γμ−1+120μ−4log2130+154γμ−1+19μ−1log2154.


## 3.2.3. Sum Connectivity Gourava Index and Entropy ofAMμγ

(41)SGVH2=∑γμ∈EH11ψγ+ψμ+ψγ×ψμ=11+3+1×3γμ−1+19μ+12+2+2×2γμ−1+22μ+12+3+2×36γμ−1+120μ−4+13+3+3×3γμ−1+19μ−1=28.058610μ+2.798766γμ−1−1.464334,ESGVH2=log228.058610γ+2.798766μγ−1−1.464334−128.058610γ+2.798766μγ−1−1.464334×17γμ−1+19μlog217+18γμ−1+22μlog218+1116γμ−1+120μ−4log2111+115γμ−1+19μ−1log2115.


## 3.2.4. First and Second Hyper-Gourava Entropies ofAMμγ

First and second hyper-Gourava indices with the help of Table4 are(42)HGO1H2=∑bw∈EH2ψb+ψw+ψb×ψw2=1+3+1×32γμ−1+19μ+2+2+2×22γμ−1+22μ+2+3+2×326γμ−1+120μ−4+3+3+3×32γμ−1+19μ−1=10494μ+1064γμ−1−709,HGO2H2=∑bw∈EH2ψb+ψw×ψbψw2=1+3×1×32γμ−1+19μ+2+2×2×22γμ−1+22μ+2+3×2×326γμ−1+120μ−4+3+3×3×32γμ−1+19μ−1=4340μ+218γμ−1−136.Entropy of first and second hyper-Gourava Indices is computed as(43)EHGO1H2=ln10494μ+1064γμ−1−709ln2−110494μ+1064γμ−1−709×4608μ+384γμ−1+989μ+γμ−1ln7ln2+24260μ+6γμ−1−4ln11ln2+4509μ+γμ−1−1ln15ln2,EHGO2H2=ln4340μ+218γμ−1−136ln2−14340μ+218γμ−1−136×2624μ+128γμ−1+50120μ+6γμ−1−4ln5ln2+7219μ+γμ−1−1ln6ln2.


## 4. Applications and Discussion of Computed Results

We calculated information entropy for two natural polymer networks, namely, the glycogenGLμγ network and amylopectin AMμγ network. The information entropies that are measured in this research are K^ Banhatti and Gourava entropies. These entropies are calculated corresponding to different K^ Banhatti and Gourava indices, namely, first and second K^ Banhatti indices, first and second K^-hyper Banhatti indices, harmonic Banhatti index, sum connectivity Banhatti index, first and second Gourava indices, sum and product connectivity Gourava indices, and first and secon hyper Gourava indices. Analytical results are obtained for these entropies. We analysed the analytical data for various parameters μ and γ, and this analysis is given in Tables 5–12. Graphical comparison is shown in Figures 6–16. Numerical analysis is performed by using MATLAB software, while Maple software is used for 3D graphical comparisons.Table 5
Numerical analysis ofK^ Banhatti indices for GLμγ.
γ,μB11B21HB21HB222,22042274411182220803,33380454818534366844,44910661226946534125,56632893636418722646,685461152046950932407,7106521436458542116340γ,μmB1−1mB2−1HbSB1−1/22,276.028 665.055 6152.057 1170.353 03,3125.476 2107.222 2250.952 4281.358 24,4181.961 9155.388 9363.923 8408.189 95,5245.485 7209.555 6490.971 4550.848 26,6316.047 6269.722 2632.095 2709.333 07,7393.647 0335.888 89787.295 2883.644 3Table 6
Numerical analysis ofK^ Banhatti indices for AMμγ.
γ,μB11B21B21HB222,23962534421782433203,36260844834434685444,487501181248146958925,51143215436629181253646,61430619320787501569607,7173722346495642190680γ,μmB1−1mB2−1HbSB1−1/22,2146.409 5125.055 6292.819 0328.618 33,3231.047 6197.222 2462.095 2518.756 24,4322.723 8275.388 9645.447 6724.720 65,5421.438 1359.555 6842.876 2946.511 56,6527.190 5449.722 21054.381 01184.128 97,7639.980 9545.888 91279.961 9437.572 8Table 7
Numerical analysis ofK^ Banhatti entropies for GLμγ.
γ,μEB11EB21EHB21EHB222,27.573 87.524 17.524 87.364 63,38.298 88.249 38.249 98.090 34,48.836 38.786 88.787 58.628 15,59.269 29.219 79.220 49.061 16,69.634 39.584 99.585 69.426 47,79.951 69.902 19.902 99.743 8γ,μEmB1−1EmB2−1EHbESB1−1/22,27.569 37.479 27.569 37.587 03,38.294 28.203 98.294 28.311 94,48.831 78.741 18.831 78.849 45,59.264 59.173 69.264 59.282 26,69.269 69.538 59.629 69.647 47,79.946 99.855 69.946 99.964 6Table 8
Numerical analysis ofK^ Banhatti entropies for AMμγ.
γ,μEB11EB21EHB21EHB222,28.524 48.474 88.475 68.316 23,39.183 59.134 09.134 98.975 64,49.666 19.616 79.617 59.458 45,510.051 610.002 110.002 99.843 96,610.374 810.325 410.326 210.167 37,710.654 810.605 310.606 210.447 3γ,μEmB1−1EmB2−1EHbESB1−1/22,28.519 78.427 48.394 18.537 53,39.178 89.086 59.059 49.196 64,49.661 49.569 19.547 59.679 25,510.046 89.954 49.937 710.064 66,610.370 010.277 610.265 410.387 97,710.649 910.557 410.549 410.667 8Table 9
Numerical analysis of Gourava indices forGLμγ.
γ,μGO1GO2SPG2,22041552238.227883,33379915663.085514,449091331491.483485,5663117996123.421796,6854523202158.900447,71065128932197.91943γ,μHGO1HGO2SG2,22240718014060.25053,33715729961699.50414,454035436524144.35535,573041590864194.80416,694175762636250.85047,7117437951840312.4942Table 10
Numerical analysis of Gourava indices forAMμγ.
γ,μGO1GO2SPG2,239611076273.63123,3625917016116.19064,4874923794162.29025,51143131096211.93026,61430538922265.11067,71737147272321.8313γ,μHGO1HGO2SG2,243687165256116.24803,369077561096183.50054,496595785164256.35055,51262411026664334.79806,61580151285596418.84307,71919171561960508.4856Table 11
Numerical analysis of Gourava entropies forGLμγ.
γ,μE(Go1)E(Go2)E(SG)2,27.56477.58277.58493,38.28968.30768.30984,48.82718.84518.8473[5, 5]9.25999.27809.2801[6, 6]9.62509.64319.6452[7, 7]9.94229.96049.9624γ,μE(HGo1)E(HGo2)E(PG)[2, 2]7.487311.59598.0322[3, 3]8.212312.3098.7589[4, 4]8.749712.83888.8523[5, 5]9.182413.26559.2852[6, 6]9.547413.62169.6503[7, 7]9.864613.937910.4138Table 12
Numerical analysis of Gourava entropies forAMμγ
γ,μEGo1EGo2ESG2,28.51498.53378.53533,39.17409.19379.19444,49.65669.67509.67715,510.042010.060410.06256,610.365210.383610.38367,710.645110.663610.6656γ,μEHGo1EHGo2EPG2,27.58498.50508.54053,38.30989.16429.19964,48.84739.646810.06765,59.280110.06766,69.645210.356010.38577,710.6708Figure 6
Graphical illustration ofK^ Banhatti indices for GLμγ. (a) The first and the second K^ Banhatti indices as well as K^-hyper Banhatti indices. (b) The sum connectivity Banhatti index, harmonic K^ Banhatti index, and modified first and second Banhatti indices.
(a)(b)Figure 7
Graphical illustration ofK^ Banhatti indices for AMμγ. (a) The first and the second K^ Banhatti indices as well as K^-hyper Banhatti indices. (b) The sum connectivity Banhatti index, harmonic K^ Banhatti index, and modified first and second Banhatti indices.
(a)(b)Figure 8
Graphical comparison between (a) modified first Banhatti entropies forGLμγ and AMμγ  and (b) modified second Banhatti entropies for GLμγ and AMμγ
(a)(b)Figure 9
Graphical comparison between (a) firstK^ Banhatti entropies for GLμγ and AMμγ and (b) second K^ Banhatti entropies for GLμγ and AMμγ.
(a)(b)Figure 10
Graphical comparison between (a) firstK^-hyper Banhatti entropies for GLμγ and AMμγ and (b) second K^-hyper Banhatti entropies for GLμγ and AMμγ.
(a)(b)Figure 11
Graphical comparison between (a) harmonicK^ Banhatti entropies for GLμγ and AMμγ and (b) sum connectivity Banhatti entropies for GLμγ and AMμγ.
(a)(b)Figure 12
Graphical illustration of Gourava indices forGLμγ. (a) The first and the second Gourava indices and Gourava hyper-indices. (b) The sum and the product Gourava connectivity indices.
(a)(b)Figure 13
Graphical illustration of Gourava indices forAMμγ. (a) The first and the second Gourava indices and Gourava hyper-indices. (b) The sum and the product Gourava connectivity indices.
(a)(b)Figure 14
Graphical comparison between (a) the first Gourava entropies forGLμγ and AMμγ and (b) the second Gourava entropies for GLμγ and AMμγ.
(a)(b)Figure 15
Graphical comparison between (a) the first hyper-Gourava entropies forGLμγ and AMμγ and (b) the second hyper Gourava entropies for GLμγ and AMμγ.
(a)(b)Figure 16
Graphical comparison between (a) the product connectivity Gourava entropies forGLμγ and AMμγ and (b) the sum connectivity Gourava entropies for GLμγ and AMμγ.
(a)(b)We can observe from Tables5, 6, 9, and 10 that when the number of monomers is the same, ΓGLμγ≤ΓAMμγ, where ΓGLμγ and ΓAMμγ represent the aforementioned topological indices for the glycogen GLμγ network and amylopectin AMμγ network, respectively. Furthermore, all of the graphical illustrations for glycogen and amylopectin are adjacent, which could be owing to the presence of monomer α-glucose, digestibility, solubility, and their utilization as energy storage in animal organs and plants, correspondingly. In addition, the findings in this section could be used in QSPR/QSAR research to predict the biological features of the natural polymers under consideration.It can be observed from Tables7, 8, 11, and 12 that when the number of monomers is the same, EΓGLμγ≤EΓAMμγ, where EΓGLμγ and EΓAMμγ represent the aforementioned information entropies corresponding to K^ Banhatti and Gourava indices for the glycogen GLμγ network and amylopectin AMμγ network, respectively. For both the structures, the sum connectivity Banhatti entropy is expanding faster than other entropies, while second K^-hyper Banhatti entropy is growing slower than other entropies. The graphical illustration shows that the entropies for glycogen and amylopectin go along the side.


## 5. Conclusion

In this study, we looked into natural polymers of polysaccharides (glycogen and amylopectin) with fascinating pharmacological applications and unique molecular structures. Nutritional activities, including energy storage for metabolism, are the principal biological functions of these polysaccharides (amylopectin and glycogen). We computedK^ Banhatti and Gourava indices and their corresponding entropies. These indices, together with entropies, are quantitatively and graphically compared. The findings of this article could be used in QSPR/QSAR analysis to predict the biological properties of the natural polymers being studied. Polysaccharide research can be expanded to include molecular graphs of other natural polymers (proteins and nucleotides) in the future.



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