# Various Uses of Topological Invariants in Jahangir GraphJβ,α

Authors: Imran Siddique; G. Muhiuddin; Abid Mahboob; Muhammad Waheed Rasheed
Journal: Journal of Mathematics
 (2022)

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## Abstract

Topological indices are analytical indicators of a graph’s topology. Graph theory has diverse applications in chemistry and conducts research on molecular structures, and its popularity has steadily increased since then. Many physicochemical attributes of a molecular substance can be identified using topological indices. Topological indices based on these chemical molecular structures can assist researchers in better understanding the physical properties, chemical reactivity, and biological activity. It is a way to fulfill the lack of experiments and give a theoretical basis for the manufacture of many chemical products. The objective of this research is to discuss bidistance degree-based topological invariants in the Jahangir graph.


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## Body

## 1. Introduction

Chemical graph theory is a field of mathematical chemistry that uses graph theory to model chemical events numerically. In the computational and theoretical aspects of chemistry, topological indices play a critical role in estimating mechanical characteristics [1–7]. In chemistry, a lot of algebraic polynomials are helpful [8, 9]. Graph theory analyses the different properties of graphs [10]. The topological indices in graphs exhibit the graphical structure as well as a variety of other aspects. They frequently rely on the distances between the vertices, the degrees of the vertices, or the network portrayed by a matrix. Many networks’ topological indices are crucial in computer-aided modeling, mathematical chemistry, and medicinal research.Weiner [11] effectively brought the theory of topological indices based on graph distance in chemistry in 1947. To correlate aspects of aromatic aldehydes with the geometries of their molecular graphs, he proposed a distance-based topological index known as the wiener index.Weiner was the first to provide a topological index assessment for graphG in 1947 [12, 13]. He created the first index, the Weiner index, and since then, many further indices have been discovered. Few of them are discussed in the investigation of physicochemical properties of various substances in G. A-index demonstrates a high degree of predictability [14]. Kulli invented the K-Banhatti and K-hyper-Banhatti indices in 2016 [15]. In 2017, Kulli proposed new further topological indices, the modified K-Banhatti index and the harmonic K-Banhatti index [16]. The Banhatti indices are to some extent inspired by the Zagreb indices given by Gutman in 1972.Baig et al. examine Revan and hyper-Revan indices for octahedral networks [17], [18]. Recently, Zhao et al. discussed the B-indices, R-index, and hyper-R-index for the very important structure of silicon carbide isomer [19]. As hydrocarbons are the main part of organic chemistry, Kulli et al. studied the multiplicative version of these indices in chemistry [20]. Revan polynomials are also introduced to get more information about the molecular structure of benzene [21]. Due to great importance of the Revan index, different forms of it were introduced, like Revan weighted Szeged index was introduced by Kandan in 2018 [22]. More advanced information about graphs and topological indices is discussed in references [23–34].


## 2. Jahangir GraphJβ,α

The Jahangir graph, abbreviated asJβ,α, is a graph with βα + 1 vertices and α (β + 1) edges ∀β ≥ 2 and α ≥ 3. Jβ,α is made up of a cycle Cβ,α plus one additional vertex that is at a distance from the vertices of Cβ,α. We will explore the different topological indices of the Jahangir graph Jβ,α. Our analyses will give a wide range of information about the Jahangir graph Jβ,α.


## 3. Preliminaries

First and second K-Banhatti indices: Kulli established the first and second K-Banhatti indices in [35] as follows:(1)Ɓ1Jβ,α=∑ue∈EGdu+de,Ɓ2Jβ,α=∑ue∈EGdu×de.Modified first and second K-Banhatti indices are as follows:(2)Ɓɱ1Jβ,α=∑ue∈EG1du+de,Ɓɱ2Jβ,α=∑ue∈EG1du×de.First and second K-hyper-Banhatti indices are as follows:(3)HƁ1Jβ,α=∑ue∈EGdu+de2,HƁ2Jβ,α=∑ue∈EGdu×de2.First and second hyper-Revan indices are as follows:(4)HR1Jβ,α=∑uv∈EGrGu+rGv2,HR2Jβ,α=∑uv∈EGrGu×rGv2.First and second Revan indices are as follows:(5)R1Jβ,α=∑uv∈EGrGu+rGv,R2Jβ,α=∑uv∈EGrGurGv.Third and F-Revan indices are as follows:(6)R⌣3Jβ,α=∑uv∈EGrGu−rGv,FRJβ,α=∑uv∈EGrGu2−rGv2.Modified first and second Revan indices are as follows:(7)Rα1Jβ,α=∑uv∈EG1rGu+rGv,Rα2Jβ,α=∑uv∈EG1rGurGv.Sum and product connectivity Revan indices are as follows:(8)SRJβ,α=∑uv∈EG1rGu+rGv,PRJβ,α=∑uv∈EG1rGurGv.General first and second Revan indices are as follows:(9)R1ϑJβ,α=∑uv∈EGrGu+rGv2,R2ϑJβ,α=∑uv∈EGrGu×rGv2.Harmonic and symmetric division Revan indices are as follows:(10)HRJβ,α=∑uv∈EG2rGu+rGv,SDRJβ,α=∑uv∈EGrGurGv+rGvrGu.Throughout this study, we will use the finite connected graphG, as shown in Figure 1. We denoted the vertices set by V (G) and the edges set by E (G). Suppose u,v∈VG, then we denote the degree of u by d (u), the distance between u and v by du,v, an edge between the vertices u and v by e = uv, the degree of an edge e by de, where de = du + dv – 2, and the maximum and minimum degrees in a graph by ΔG and δG respectively. The nodes and lines are also called the elements of a graph.Figure 1
Jahangir graphs.


## 4. Bidistance Edges

We have introduced a new way to find the TIs. It is an approach named bi-distance edges. In bi-distance edges, there are two single edges and three vertices. The concept is very beneficial for estimating the physio-chemical properties of chemical compounds. This concept gives very good correlation with many properties of graphs. The single-distance edge partition and the bidistance edge partition are listed in Tables1 and 2. The connected Jahangir graph J5,3 is shown in Figure 2.Table 1
The single distance edge partition.
(d (u), d (v))FrequencydGe(2, 2)92(3, 2)63(3, 3)34Table 2
The bidistance edge partition.
(d (u), d (v))FrequencydGe(2, 2)92(3, 2)123(3, 3)34Figure 2
Jahangir graphJ5,3.The single distance Banhatti index for Jahangir graphJ5,3 is given as follows:(11)Ɓ1J5,3=∑ue∈EGdu+de=92+2+63+3+33+4=36+36+21=93.The bidistance first Banhatti index for the Jahangir graphJ5,3 is computed as follows:(12)Ɓ1J5,3=∑ue∈EGdu+de=92+2+123+3+33+4=36+72+21=93.


## 5. Main Results

In this section, we study twenty important results related to TIs and also check out their results in graph form. First, we have edges and vertex divisions, as given in Table3. The edges are partitioned into different categories on the basis of degree. In Table 3, we have Revan vertices rGu,rGv. The Revan indices are defined as follows:(13)rGu=ΔG+δG−dGu,rGv=ΔG+δG−dGv,where terms ΔG and δG are the maximum and minimum degrees among the vertices respectively, and the degree of Banhatti edge is computed as follows:(14)de=dGu+dGv−2.Table 3
Degree-based partition of edges.
d (u), d (v)Frequencyd (e)rG (u)rG (v)(3, 2)2α3α − 1α(2, 2)αβ − 2β2αα(3, 3)(1/2)α2 − (1/2) α4α − 1α − 1(2,α)2ααα2α + 2We are dealing with the division of edges and various terms derived from this division. The Jahangir graph has four types of different edge parcels, as listed in Table3.Theorem 1.
ConsiderJβ,α represents the Jahangir graph, then(15)Ɓ1Jβ,α=8αβ+13α2+19α−16β,Ɓ2Jβ,α=8αβ+16α2+2α3+18α−16β.Proof.
LetJβ,α be the Jahangir graph. The first K-Banhatti index of J [β, α] is then calculated by using Table 3:(16)Ɓ1Jβ,α=∑ue∈EGdu+de=∑uv=edu+de+dv+de=2α3+3+2+3+αβ−2β2+2+2+2+12α2−12α3+4+3+4+2α2+α+α+α=2α11+αβ−2β8+12α2−12α14+2α2+3α=8αβ+132α2+19α−16β.
Second K-Banhatti index ofJ [β, α] is determined as follows:(17)Ɓ2Jβ,α=∑ue∈EGdu×dv=∑uv=edu×de+dv×de=2α3×3+2×3+αβ−2β2×2+2×2+12α2−12α3×4+3×4+2α2×α+α×α=2α15+αβ−2β8+12α2−12α24+2α2α+α2=8αβ+16α2+2α3+18α−16β.Theorem 2.
ConsiderJβ,α represents the Jahangir graph, then(18)Ɓɱ1Jβ,α=12αβ+17α2+62105α−β+3α+22+α,Ɓɱ2Jβ,α=12αβ+112α2+136α−2αβ+1.Proof.
LetJβ,α be the Jahangir graph. The modified first K-Banhatti index of J [β, α] is then determined using Table 3 as follows:(19)Ɓɱ1Jβ,α=∑ue∈EG1du+de=∑uv=e1du+de+1dv+de=2α16+15+αβ−2β14+14+12α2−12α17+17+2α12+α+12α=2α1130+αβ−2β12+12α2−12α27+2α3α+24α+2α2=12αβ+17α2+62105α−β+3α+22+α.
From the definition of second K-Banhatti index (Ɓɱ2 (Jβ,α)), we have(20)Ɓɱ2Jβ,α=∑ue∈EG1du×de=∑uv=e1du×de+1dv×de=2α19+16+αβ−2β14+14+12α2−12α112+112+2α12α+1α2=12αβ+112α2+136α+2α−β+1.Theorem 3.
ConsiderJβ,α represents the Jahangir graph, then(21)HƁ1Jβ,α=32αβ+10α3+57α2+81α−64β,HƁ2Jβ,α=32αβ+2α5+8α3+144α2+90α−64β.Proof.
LetJβ,α be the Jahangir graph. The first K-hyper-Banhatti index of J [β, α] is evaluated using Table 3:(22)HƁ1Jβ,α=∑uedu+de2=∑uv=e∈Edu+de2+dv+de2=2α3+32+2+32+αβ−2β2+22+2+22+12α2−12α3+42+3+42+2α2+α2+2α2=32αβ+10α3+57α2+81α−64β.
From the definition of second K-hyper-Banhatti indexHƁ2 (Jβ,α), we have(23)HƁ2Jβ,α=∑uedu×dv2=∑uv=e∈Edu×de2+dv×de2=2α3×32+2×32+αβ−2β2×22+2×22+12α2−12α3×42+3×42+2α2×α2+α×α2=32αβ+2α5+8α3+144α2+90α−64β.Theorem 4.
ConsiderJβ,α represents the Jahangir graph, then(24)HR1Jβ,α=2α4+20α3+22α2+8α+4α3β−8α2β,HR2Jβ,α=4α5+8α4+16α3+8α+2α3β−4α2β.Proof.
LetJβ,α be the Jahangir graph. The first hyper-Revan index of J [β, α] is then calculated using Table 3 as follows:(25)HR1Jβ,α=∑uv∈EGrGu+rGv2=2αα−1+α2+αβ−2βα+α2+12α2−12αα−1+α−12+2αα+2α+22=2α2α−12+αβ−2β2α2+12α2−12α2α−22+2αα+2α+22=2α4α2−4α+1+αβ−2β4α2+12α2−12α4α2−8α+42+2α9α2+12α+4=2α4+20α3+22α2+8α+4α3β−8α2β.
By the definition of second hyper-Revan index (HR2 (Jβ,α)), we have(26)HR2Jβ,α=∑uv∈EGrGu×rGv2=2α2α×12+αβ−2β2α2+12α2−12α2α×22+2αα×2α+22=2α2α2+αβ−2β2α2+12α2−12α4α2+2α2α2+22=4α5+8α4+16α3+8α+2α3β−4α2β.Theorem 5.
ConsiderJβ,α represents the Jahangir graph, then(27)R⌣3Jβ,α=3α2+5α,FRJβ,α=2α3β−4α2β+11α3+16α2+9α.Proof.
LetJβ,α be the Jahangir graph. The Revan index of J [β, α] is then calculated using Table 3 as follows:(28)R⌣3Jβ,α=∑uv∈EGrGu−rGv,=2α1+αβ−2β0+12α2−12α2+2αα+2=2α+α2−α+2α2+4α=3α2+5α.
By the definition of F-Revan index (FR (Jβ,α)), we have(29)FRJβ,α=∑uv∈EGrGu2−rGv2=2αα−12+α2+αβ−2βα2+α2+12α2−12αα−12+α−12+2αα2+2α+22=2αα2+1−2α+α2+αβ−2β2α2+12α2−12αα2+1−2α+α2+1−2α+2αα2+4α2+4+8α=2α2α2−2α+1+αβ−2β2α2+12α2−12α2α2−4α+22+2α5α2+8α+4=2α3β−4α2β+11α3+16α2+9α.Theorem 6.
ConsiderJβ,α represents the Jahangir graph, then(30)R1Jβ,α=2α2β−2αβ+α3+8α2+3α,R2Jβ,α=α3β−2α2β+12α4+92α3+72α2−12α.Proof.
LetJβ,α be the Jahangir graph. The first Revan index of J [β, α] is then determined using Table 3 as follows:(31)R1Jβ,α=∑uv∈GrGu+rGv=2αα−1+α+αβ−2βα+α+12α2−12αα−1+α−1+2αα+2α+2=2α2α−1+αβ−2β2α2+12α2−12α2α−2+2α3α+2=2α2β−2αβ+α3+8α2+3α.
From the definition of second Revan index (R2 (Jβ,α)), we have(32)R2Jβ,α=∑uv∈GrGurGv=2αα−1×α+αβ−2βα×α+12α2−12αα−1×α−1+2αα×2α+2=2αα2−α+αβ−2βα2+12α2−12αα2−2α+1+2α2α2+2α=α3β−2α2β+12α4+92α3+72α2+12α.Theorem 7.
ConsiderJβ,α represents the Jahangir graph, then(33)Rα1Jβ,α=2α2α−1+αβ−2β2α+α24α−1−α4α−1+2α3α+2,Rα2Jβ,α=2α−1+αβ−2βα2+α221−2α+α2−α21−2α+α2+α1+α2.Proof.
LetJβ,α be the Jahangir graph. The modified first Revan index of J [β, α] is then calculated using Table 3 as follows:(34)Rα1Jβ,α=∑uv∈EGrGu+rGv=2α1α−1+α+αβ−2β1α+α+12α2−12α1α−1+α−1+2α1α+2α+2=2α12α−1+αβ−2β12α+12α2−12α12α−2+2α13α+2=2α2α−1+αβ−2β2α+α24α−1−α4α−1+2α3α+2.
From the definition of modified second Revan index (αR2 (Jβ,α)), we have(35)Rα2Jβ,α=∑uv∈EG1rGurGv=2α1α−1α+αβ−2β12α2+12α2−12α1α−12+2α1α2α+2=2α1−α+α2+αβ−2β1α2+12α2−12α11−2α+α2+2α12+2α2=2α−1+αβ−2βα2+α221−2α+α2−α21−2α+α2+α1+α2.Theorem 8.
ConsiderJβ,α represents the Jahangir graph, then(36)SRJβ,α=2α2α−1+αβ−2β2α+α222α−1−α22α−1+2α3α+2,PRJβ,α=2α1αα−1+β−2βα+α22α−1−α2α−1+2α1α2α+2.Proof.
LetJβ,α be the Jahangir graph. The sum connectivity Revan index of J [β, α] is then calculated using Table 3 as follows:(37)SRJβ,α=∑uv∈EG1rGu+rGv=2α1α−1+α+αβ−2β1α+α+12α2−12α1α−1+α−1+2α1α+2α+2=2α2α−1+αβ−2β2α+α222α−1−α22α−1+2α3α+2.
From the definition of product connectivity Revan index (PR (Jβ,α)), we have(38)PRJβ,α=∑uv∈EG1rGurGv=2α1αα−1+αβ−2β1α2+12α2−12α1α−12+2α1α2α+2=2α1αα−1+β−2βα+α22α−1−α2α−1+2α1α2α+2.Theorem 9.
ConsiderJβ,α represents the Jahangir graph, then(39)R1ϑJβ,α=2α2β−2αβ+α3+8α2+3α,R2ϑJβ,α=α5β−2βα4+12α6+232α5+5α4+3α3−32α2−12α.Proof.
LetJβ,α be the Jahangir graph. The general first Revan index of J [β, α] is then calculated using Table 3 as follows:(40)R1ϑJβ,α=∑uv∈EGrGu+rGv2=2αα−1+αϑ+αβ−2βα+αϑ+12α2−12αα−1+α−1ϑ+2αα+2α+2ϑ=2α2α−1ϑ+αβ−2β2αϑ+12α2−12α2α−2ϑ+2α3α+2ϑ∴ϑ=2=2α2β−2αβ+α3+8α2+3α.
From the definition of general second Revan index (R2ϑ (Jβ,α)), we have(41)R2ϑJβ,α=∑uv∈EGrGu×rGv2=2αα−1αϑ+αβ−2βα2ϑ+12α2−12αα−12ϑ+2α2α+2αϑ=2αα2−αϑ+αβ−2βα2ϑ+12α2−12αα−12ϑ+2α2α2+2αϑ∴ϑ=2=2αα2−α2+αβ−2βα22+12α2−12αα−122+2α2α2+2α2=2αα4+α2−2α3+α5β−2βα4+12α2−12αα4+4α3+6α2+4α+1+2α4α4+α2+4α3=α5β−2βα4+12α6+232α5+5α4+3α3−32α2−12α.Theorem 10.
ConsiderJβ,α represents the Jahangir graph, then(42)HRJβ,α=4α2α−1+β−2βα+α22α−1−α2α−1+4α3α+2,SDRJβ,α=22α2+1−2αα−1+2αβ−4β+α2−α+5α2+8α+4α+1.Proof.
LetJβ,α be the Jahangir graph. The harmonic Revan index of J [β, α] is then calculated using Table 3 as follows:(43)HRJβ,α=∑uv∈EG2rGu+rGv=2α2α−1+α+αβ−2β22α+12α2−12α22α−2+2α2α+2α+2=4α2α−1+β−2βα+12α2−12α1α−1+4α3α+2=4α2α−1+β−2βα+α22α−1−α2α−1+4α3α+2.
From the definition of symmetric division Revan index (SDR (Jβ,α)), we have(44)SDRJβ,α=∑uv∈EGrGurGv+rGvrGu=2αα−1α+αα−1+αβ−2βαα+αα+12α2−12αα−1α−1+α−1α−1+2αα2α+2+2α+2α=2αα2+α−12αα−1+αβ−2β2+12α2−12α2+2αα2+2α+222α+2α=22α2+1−2αα−1+2αβ−4β+α2−α+5α2+8α+4α+1.


## 6. Discussion and Graphical Representation

Graphs are a type of perspective drawing that uses lines and points to depict a precise sequence of data and information. Network theory, molecular chemistry, and many disciplines of mathematics, as well as medicines and organic chemistry, all utilize it. In this section, we put the value ofβ = 2, 3, 4, 5, 6 and α = 3, 4, 5, 6, 7. Then, we evaluate all the values of β and α in the topological indices, which are discussed above. The numerical representation of the above calculated topological indices is listed in Table 4 and the graphical representations are shown in Figures 3 and 4.Table 4
Numerical comparison.
Banhatti indices(Jβ,α) = (2, 3)(Jβ,α) = (3, 4)(Jβ,α) = (4, 5)(Jβ,α) = (5, 6)(Jβ,α) = (6, 7)Ɓ1 (Jβ,α)1903325167421010Ɓ2 (Jβ,α)26850483612761836ɱƁ1 (Jβ,α)6.25679.980614.951621.185028.689ɱƁ2 (Jβ,α)3.50035.9449.621914.499721.1344HƁ1 (Jβ,α)10902068346453387686HƁ2 (Jβ,α)23325416116842364445004HR1 (Jβ,α)99625605541063218676HR2 (Jβ,α)2112739220140464169492Ř3 (Jβ,α)4268100138182FR (Jβ,α)5041188242044467560R1 (Jβ,α)1322765008221260R2 (Jβ,α)192566126024634382αR1 (Jβ,α)2.8293.4644.1494.8585.578αR2 (Jβ,α)2.2721.9441.7971.7181.669SR (Jβ,α)6.8099.73313.08916.81820.882PR (Jβ,α)5.84097.07438.42719.833511.2688R1ϑ (Jβ,α)1322765008221260R2ϑ (Jβ,α)3792168065471145743337092HR (Jβ,α)5.65766.64298.29879.715211.1570SDR (Jβ,α)91.334161.134249.917358.772487.667Figure 3
Comparison of topological indices.Figure 4
Comparison of topological indices.


## 7. Applications

The Banhatti and Revan indices have a strong relationship with the physical and chemical characteristics of various structures. Asthma drugs are also useful for the treatment of COVID-19. The first Banhatti index has a correlation coefficientr = 0.974 with molar volume, and the second Banhatti index is correlated with the boiling point (r = 0.9192) and enthalpy (r = 0.9125) of these drugs. Hyper-Revan indices are effective to describe the molar volume of asthma drugs. Both these indices have a correlation with the different properties of silicon carbide structures discussed in this manuscript. These indices also tell us about the structural properties of linear alkanes.


## 8. Conclusion

In this article, we discuss some topological invariants such as Ɓ1 (Jβ,α), Ɓ2 (Jβ,α), ɱƁ1 (Jβ,α), ɱƁ2 (Jβ,α), HB1 (Jβ,α), HB2 (Jβ,α), HR1 (Jβ,α), HR2 (Jβ,α), Ř3 (Jβ,α), FR (Jβ,α), R1 (Jβ,α), R2 (Jβ,α), αR1 (Jβ,α), αR2 (Jβ,α), SR (Jβ,α), PR (Jβ,α), R1ϑ (Jβ,α), R2ϑ (Jβ,α), HR (Jβ,α), and SDR (Jβ,α) for the Jahangir graph. These topological indices aid in the decoding of the graph’s hidden information storage. Isomorphism does not affect these results. These findings could have a significant impact on industrial applications and pharmaceuticals.



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