# On Generalized Topological Indices for Some Special Graphs

Authors: Sheeba Afridi; Muhammad Yasin Khan; Gohar ALi
Journal: Journal of Mathematics
 (2022)

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## Abstract

Topological indices are numeric values associated with a graph and characterize its structure. There are various topological indices in graph theory such as degree-based, distance-based, and counting-related topological indices. Among these indices, degree-based indices are very interesting and studied well in literature. In this work, we studied the generalized form of harmonic, geometric-arithmetic, Kulli–Basava indices, and generalized power-sum-connectivity index for special graph that are bridge graph over path, bridge graph over cycle, bridge graph over complete graph, wheel graph, gear graph, helm graph, and square lattice graph. We found exact values for the stated indices and for the stated special graphs. We also investigated the generalized form of the indices for various properties of alkane isomers, from which we obtained interesting results which are closed to that of experimental obtained results.


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## Body

## 1. Introduction

Graph theory, as applied in the field of molecular structures, representing an interdisciplinary science called molecular topology or chemical graph theory. By means of graph theory, statistics, and set theory, we identify structural features involved in structure property activity relationships. Modeling unknown structures and molecules can be classified by means of characterization of chemical structure topologically having the desired properties. In the last few decades, a lot of research studies have been conducted. Numerical value associated with chemical constitutions of chemical structures having many physicochemical properties, biological activity, or chemical reactivity. Topological indices are introduced on the basis of transforming molecular graph into a numeric value which characterizes the topology of a molecular graph. Molecular compound and molecules are often modeled by molecular graphs. For characterization of chemical compound, chemical graph is a model in this regard. A molecular graph is a simple graph where vertices are used for atoms and edges are for bonds. This can be described in many ways like that of by drawing, by means of polynomial, by means of a sequence of numbers, by means of a matrix, or by a derived number which is called topological index. A numeric value associated with a graph is termed as topological index. Some of the well-known topological indices are degree-based, counting-related, and distance based topological indices. The generalized harmonic index, geometric arithmetic index, Kulli–Basava indices, and generalized power-sum-connectivity index are of the most important. For two isomorphic graphH and G, we have that TopH=TopG, where Top is a topological index. The number of edges and vertices of a graph are topological indices [1–5]. Let G=V,E be a simple graph having vertex and edge set VG and EG, respectively. For any v∈VG, the neighbors of v are denoted by Nv and are defined as Nv=u∈VG|vu∈EG. We denote the degree of a vertex u by du or du and Su=∑v∈NGudv is the degree of neighbors of u. VG and EG are the number of vertices and edges of G, respectively. The graph denoted by Kn is called complete graph if all the vertices are connected by an edge [6–8].In [9], the authors computed ABC, ABC4, GA, and GA5 indices for special graphs that are Cayley tree, square lattice, and complete bipartite graph. In [10], the authors determined hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index, Zagreb polynomials, and M-polynomial for nanostructures of bridge graphs. In [11], the authors introduced Kulli–Basava indices and studied their chemical and mathematical properties that have good response in isomers degeneracy. Also, the authors obtained closed results for the stated indices for some families of graphs and established relations for connecting these indices with other existing indices which are in literature already. In addition, they have found the Kulli–Basava indices for some graph operations. In [12], the author proposed first hyper-Kulli–Basava index and second hyper-Kulli–Basava index and their polynomials of graphs and computed exact results for complete graph, gear graph, and helm graph. Also, the author determined the Kulli–Basava indices of gear graphs and helm graphs. In [13], the authors introduced symmetric division Kulli–Basava index, first and second multiplicative Kulli–Gourava indices, multiplicative F1-Kulli–Basava index, and multiplicative a,b-Kulli–Basava index of graph. The author computed theses indices for regular graph, wheel graph, gear graph, and helm graph. In [14], the authors introduced multiplicative sum-connectivity Kulli–Basava index, multiplicative product connectivity Kulli–Basava index, multiplicative ABC Kulli–Basava index, and multiplicative GA Kulli–Basava index of graphs. The authors computed the multiplicative connectivity Kulli–Basava indices of regular graph, wheel graph, and helm graph. In [15], the authors introduced multiplicative a,b-status index for graph. The authors found exact expressions for the multiplicative a,b-status index for friendship graphs and wheel graphs. In [16], the author obtained expression for Harmonic index and Randic index of generalized transformation graph Gxy and their complement.In this study, we generalized geometric-arithmetic index and Kulli–Basava indices. Also, we found exact values of generalized harmonic index, generalized geometric-arithmetic index, generalized Kulli–Basava indices, and generalized power-sum-connectivity index for some special graphs that are bridge graph over path, bridge graph over cycle, bridge graph over complete graph, wheel graph, gear graph, helm graph, and square lattice graph. The generalize power-sum-connectivity index was introduced in [17]. Indices that we studied here in generalized form are the following. Generalized Harmonic index is given by(1)HξG=∑pq∈EG2dp⋅dqξ.Generalized geometric-arithmetic index is given by(2)GAξG=∑pq∈EG2dpdqξdp+dq.Generalized Kulli–Basava indices are given by(3)KB1ξG=∑rs∈EGSer+Sesξ,KB2ξG=∑rs∈EGSer⋅Sesξ.where Ser=∑e∈NerdGe and dGe=dGa+dGr−2 if e=ar. Generalized power-sum-connectivity index is given by(4)YξG=∑gs∈EGdgdg+dsdsξ.


## 2. Bridge Graph over PathPn

Theorem 1 (see [10]).
LetGmPn,u be a bridge graph over path Pn; then, the number of vertices in GmPn,u are mn and number of edges are mn−1.Theorem 2.
LetGmPn;u,n>2 be bridge graph over path Pn. Then,(5)HξGmPn;u=2m3ξ+5ξ+6ξ−3.4ξ+4ξn+22ξ+2−2ξ+1⋅3ξ+1,GAξGmPn;u=2ξ+13⋅m+22ξ−1mn−3m+2+2ξ+13ξ5⋅m+32ξ−1m−3,KB1ξGmPn;u=m⋅5ξ+m⋅7ξ+mn−4⋅8ξ+2⋅9ξ+4⋅15ξ+m−4⋅16ξ+2⋅21ξ+m−5⋅22ξ,KB2ξGmPn;u=m⋅4ξ+m⋅12ξ+mn−4⋅16ξ+2⋅20ξ+4⋅50ξ+m−4⋅55ξ+2⋅110ξ+m−5⋅121ξ,YξGmPn;u=m5ξ+31ξ+54ξ+8ξn−3+22ξ2ξ+1−32ξ+1.
The graph is given in Figure1.Figure 1
Bridge graph over pathPn.Proof.
Let a path onn vertices be Pn, and GmPn;u is the bridge graph over path Pn. Using Theorem 1, mn and mn−1 are the vertices and edges, respectively. Making partition between edges of GmPn;u in such a way that E1 consists of edges g,s, where dg=1 and ds=2, such type of edges are m in number. E2 consists of edges in which dg=2 and ds=2, such type of edges are mn−3m+2 in number which can be calculated as n−1−1+n−1−1+m−1−1n−1−2=2n−4+m−2n−3=mn−3m+2. E3 consists of edges g,s in which dg=2 and ds=3; these edges are m in numbers. Similarly, E4 consists of edges g,s, where dg=3 and ds=3, such edges are m−3 in numbers. Next, we get the following interesting results:(6)HξGmPn;u=∑gs∈E12dg+dsξ+∑gs∈E22dg+dsξ+∑gs∈E32dg+dsξ+∑gs∈E42dg+dsξ=2m3ξ+5ξ+6ξ−3⋅4ξ+4ξn+22ξ+2−2ξ+1⋅3ξ+1,GAξGmPn;u=∑gs∈E12dg⋅dsξdg+ds+∑gs∈E22dg⋅dsξdg+ds+∑gs∈E32dg⋅dsξdg+ds+∑gs∈E42dg⋅dsξdg+ds=2ξ+13⋅m+22ξ−1mn−3m+2+2ξ+1⋅3ξ5⋅m+32ξ−1m−3,KB1ξGmPn;u=∑gs∈E1Seg+Sesξ+∑gs∈E2Seg+Sesξ+∑gs∈E3Seg+Sesξ+∑gs∈E4Seg+Sesξ=m⋅5ξ+m⋅7ξ+mn−4⋅8ξ+2⋅9ξ+4⋅15ξ+m−4⋅16ξ+2⋅21ξ+m−5⋅22ξ,KB2ξGmPn;u=∑gs∈E1Seg⋅Sesξ+∑gs∈E2Seg⋅Sesξ+∑gs∈E3Seg⋅Sesξ+∑gs∈E4Seg⋅Sesξ=m⋅4ξ+m⋅12ξ+mn−4⋅16ξ+2⋅20ξ+4⋅50ξ+m−4⋅55ξ+2⋅110ξ+m−5⋅121ξ,YξGmPn;u=∑gs∈E1dgdg+dsdsξ+∑gs∈E2dgdg+dsdsξ+∑gs∈E3dgdg+dsdsξ+∑gs∈E4dgdg+dsdsξ=m5ξ+31ξ+54ξ+8ξn−3+22ξ2ξ+1−32ξ+1.


## 3. Bridge Graph over CycleCn

Theorem 3 (see [10]).
LetGmCn,u be a bridge graph over cycle Cn; then, the number of vertices in GmCn,u is mn and number of edges is mn+m−1.Theorem 4.
LetGmCn;u, n>3, be bridge graph over cycle Cn; then,(7)HξGmCn;u=m22ξ+1n−22ξ+2+2ξ+2.3ξ+23ξ+1+23⋅5ξ+22⋅7ξ−2ξ+3⋅3ξ−23ξ+1⋅3,GAξGmCn;u=m22ξ−1n−22ξ+23ξ+13+24ξ−2+2ξ+3⋅3ξ5+2aξ+2⋅3ξ7−23ξ+23+24ξ−2⋅3,KB1ξGmCn;u=4⋅9ξ+2m−210ξ+mn−48ξ+416ξ+425ξ+2m−426ξ+239ξ+m−540ξ,KB2ξGmCn;u=4⋅20ξ+2m−224ξ+mn−416ξ+455ξ+4114ξ+2m−4120ξ+2380ξ+m−5400ξ,YξGmCn;u=23ξmn−23ξ+1m+22ξ+1⋅65ξm+29ξm+22⋅31ξ+2⋅283ξ−22ξ+2⋅65ξ−29ξ⋅3.
The graph is shown in Figure2.Figure 2
Bridge graph over cycleCn.Proof..
Let cycle onn vertices be Cn and GmCn;u be the bridge graph. According to Theorem 3, mn and mn+m−1 are vertices and edges, respectively. For computing topological indices for GmCn;u over Cn, we divide the edge set EGmCn;u into five partitions which are based on the degrees of end vertices. First edge partition E1 contains mn−2m edges gs having dg=2 and ds=2. E2 consists of 4 edges gs having dg=2 and ds=3. E3 contains 2m−4 edges gs having dg=2 and ds=4. E4 contains 2 edges gs having dg=3 and ds=4. E5 contains m−3 edges gs having dr=4 and ds=4.
Now, the stated topological indices forGmCn,u are calculated in the follows:(8)HξGmCn;u=∑gs∈E12dg+dsξ+∑gs∈E22dg+dsξ+∑gs∈E32dg+dsξ+∑gs∈E42dg+dsξ+∑gs∈E52dg+dsξ=m22ξ+1n−22ξ+2+2ξ+2⋅3ξ+23ξ+1+23⋅5ξ+22⋅7ξ−2ξ+3⋅3ξ−23ξ+1⋅3,GAξGmCn;u=∑gs∈E12dg⋅dsξdg+ds+∑gs∈E22dg⋅dsξdg+ds+∑gs∈E32dg⋅dsξdg+ds+∑rs∈E42dg⋅dsξdr+ds+∑rs∈E52dg⋅dsξdr+ds=m22ξ−1n−22ξ+23ξ+13+24ξ−2+2ξ+3⋅3ξ5+2aξ+2⋅3ξ7−23ξ+23+24ξ−2⋅3,KB1ξGmCn;u=∑gs∈E1Seg+Sesξ+∑gs∈E2Seg+Sesξ+∑gs∈E3Seg+Sesξ+∑gs∈E4Seg+Sesξ+∑gs∈E5Seg+Sesξ=4⋅9ξ+2m−210ξ+mn−48ξ+416ξ+425ξ+2m−426ξ+239ξ+m−540ξ,KB2ξGmCn;u=∑gs∈E1S(eg⋅Sesξ+∑gs∈E2S(eg⋅Sesξ+∑gs∈E3S(eg⋅Sesξ+∑gs∈E4S(eg⋅Sesξ+∑gs∈E5S(eg⋅Sesξ=4⋅20ξ+2m−224ξ+mn−416ξ+455ξ+4114ξ+2m−4120ξ+2380ξ+m−5400ξ,YξGmCn;u=∑gs∈E1dgdg+dsdsξ+∑gs∈E2dgdg+dsdsξ+∑gs∈E3dgdg+dsdsξ+∑gs∈E4dgdg+dsdsξ+∑gs∈E5dgdg+dsdsξ=23ξmn−23ξ+1m+22ξ+1⋅65ξm+29ξm+22⋅31ξ+2⋅283ξ−22ξ+2⋅65ξ−29ξ⋅3.


## 4. Bridge Graph over Complete GraphKn

Theorem 5.
LetGmKn;u be bridge graph over complete graph Kn with n>2; then,(9)HξGmKn;u=222n+1ξ+2ξ+1m−3n+1ξ+22n−12n−1ξ+2ξ+1m−2n−1nξ+2ξmn−2n−1ξ+1,GAξGmKn;u=22n2+nξ2n+1+m−3n+12ξ−1+22n−1n2−nξ2n−1+m−2n−1n2−1ξn+mn−2n−12ξ2,YξGmKn;u=2nn+n+1n+1ξ+2ξm−3n+1ξn+1+2n−1n−1n−1+nnξ+m−2n−1n−1n−1+n+1n+1ξ+2ξ−1mn−2n−1ξn−1+1.KB1ξGmKn;u=2n+n−12n−3+4n−1+n−12n−2ξ+24n−1+n−12n−2+4n+2n−13ξ+m−54n+n−12n−2+4n+n−12n−1ξ+2n−22n−3+n−22n−4+4n−4+n−22n−4ξ+2n−22n−2+n−22n−4+4n−1+n−12n−2ξ+m−4n−12n−2+n−22n−4+4n+n−12n−2ξ+n−1n−2n−22n−4+2n−3+n−22n−4+2n−3ξ+m−2n−2n−12n−22n−4+2n−2+n−22n−4+2n−2ξ,KB2ξGmKn;u=2n+n−12n−3⋅4n−1+n−12n−2ξ+24n−1+n−12n−2⋅4n+2n−13ξ+m−54n+n−12n−2⋅4n+n−12n−1ξ+2n−22n−3+n−22n−4⋅4n−4+n−22n−4ξ+2n−22n−2+n−22n−4⋅4n−1+n−12n−2ξ+m−4n−12n−2+n−22n−4⋅4n+n−12n−2ξ+n−1n−2n−22n−4+2n−3⋅n−22n−4+2n−3ξ+m−2n−2n−12n−22n−4+2n−2⋅n−22n−4+2n−2ξ.
The graph is given in Figure3.Figure 3
Bridge graph over complete graphKn.Proof..
Let complete graph of ordern be Kn and GmKn,u be the bridge graph over complete graph. The bridge graph GmKn,u consists mn vertices and mnn−1+2m−2/2 edges. For computing topological indices for bridge graph GmKn;u over complete graph Kn, we divide the edges into five partitions based on degree of end vertices.
The first partitionE1 consists of 2 edges gs such that dg=n and ds=n+1; second edge partition E2 contains n−3 edges gs such that dg=n+1 and ds=n+1. The third edge partition E3 consists of 2n−2 edges gs such that dg=n−1 and ds=n. The fourth edge partition E4 consists of m−2n−1 edges gs such that dg=n−1 and ds=n+1. The last and fifth edge partition consists mn−2n−1/2 edges gs such that dg=n−1 and ds=n−1.
Now, the desired topological indices are the following:(10)HξGmKn;u=∑gs∈E12dg+dsξ+∑gs∈E22dg+dsξ+∑gs∈E32dg+dsξ+∑gs∈E42dg+dsξ+∑gs∈E52dg+dsξ=2n+n+1ξE1+2n+1+n+1ξE2+2n−1+nξE3+2n−1+n+1ξE4+2n−1+n−1ξE5=222n+1ξ+2ξ+1m−3n+1ξ+22n−12n−1ξ+2ξ+1m−2n−1nξ+2ξmn−2n−1ξ+1,GAξGmKn;u=∑gs∈E12dg⋅dsξdg+ds+∑gs∈E22dg⋅dsξdg+ds+∑gs∈E32dg⋅dsξdg+ds+∑gs∈E42dg⋅dsξdg+ds+∑gs∈E52dg⋅dsξdg+ds=22n2+nξ2n+1+m−3n+12ξ−1+22n−1n2−nξ2n−1+m−2n−1n2−1ξn+mn−2n−12ξ2,KB1ξGmKn;u=∑gs∈E1Seg+Sesξ+∑gs∈E2Seg+Sesξ+∑gs∈E3Seg+Sesξ+∑gs∈E4Seg+Sesξ+∑gs∈E5Seg+Sesξ=2n+n−12n−3+4n−1+n−12n−2ξ+24n−1+n−12n−2+4n+2n−13ξ+m−54n+n−12n−2+4n+n−12n−1ξ+2n−22n−3+n−22n−4+4n−4+n−22n−4ξ+2n−22n−2+n−22n−4+4n−1+n−12n−2ξ+m−4n−12n−2+n−22n−4+4n+n−12n−2ξ+n−1n−2n−22n−4+2n−3+n−22n−4+2n−3ξ+m−2n−2n−12n−22n−4+2n−2+n−22n−4+ξ,KB2ξGmKn;u=∑gs∈E1Seg⋅Sesξ+∑gs∈E2Seg⋅Sesξ+∑gs∈E3Seg⋅Sesξ+∑gs∈E4Seg⋅Sesξ+∑gs∈E5Seg⋅Sesξ=2n+n−12n−3⋅4n−1+n−12n−2ξ+24n−1+n−12n−2⋅4n+2n−13ξ+m−54n+n−12n−2⋅4n+n−12n−1ξ+2n−22n−3+n−22n−4⋅4n−4+n−22n−4ξ+2n−22n−2+n−22n−4⋅4n−1+n−12n−2ξ+m−4n−12n−2+n−22n−4⋅4n+n−12n−2ξ+n−1n−2n−22n−4+2n−3⋅n−22n−4+2n−3ξ+m−2n−2n−12n−22n−4+2n−2⋅n−22n−4+2n−2ξ,YξGmKn;u=∑gs∈E1dgdg+dsdsξ+∑gs∈E2dgdg+dsdsξ+∑gs∈E3dgdg+dsdsξ+∑gs∈E4dgdg+dsdsξ+∑gs∈E5dgdg+dsdsξ,=2nn+n+1n+1ξ+2ξm−3n+1ξn+1+2n−1n−1n−1+nnξ+m−2n−1n−1n−1+n+1n+1ξ+2ξ−1mn−2n−1ξn−1+1.


## 5. Square Lattice,SLn

Theorem 6.
LetSLn be square lattice with n≥1; then,(11)HξSLn=24⋅5ξ+2ξ+1⋅3ξn−3+2⋅7ξn−2+23ξ+12n2−10n+12,GAξSLn=2ξ+1⋅3ξ58+22⋅32ξ−1n−3+22ξ+3⋅3ξn−27+24ξ−22n2−10n+12,KB1ξSLn=2ξ+3⋅32ξ+0⋅2352ξ+2ξ+2n−513ξ+23ξ+2ξ+3⋅17ξ+22ξ+2⋅32ξn−4+23⋅35ξ+24ξ⋅3ξ2n2−14n+24,KB2ξSLn=23ξ+3⋅32ξ+22ξ+3⋅39ξ+n−522⋅132ξ+23⋅232ξ+23ξ+3⋅33ξ+22⋅299ξn−4+2ξ+3⋅253ξ+26ξ⋅32ξ2n2−14n+24,YξSLn=23⋅31ξ+22ξ+2⋅32ξn−3+22⋅283ξn−2+29ξ2n2−10n+12.
The graph is given in Figure4.Figure 4
Square lattice graph,SLn.Proof.
LetSLn be square lattice; then, the number ov vertices are n2 and number of edges are 2nn−1. Here, we divide the edge set into four partitions. First partition E1 consists of 8 edges gs, such that dg=2 and ds=3. Second edge partition E2 consists of 4n−3 edges gs, such that dg=3 and ds=3. The third edge partition E3 consists of 4n−2 edges gs, such that dg=3 and ds=4. The fourth edge partition E4 consists of 2n2−10n+12 edges gs, such that dg=4 and ds=4. The desired topological indices for the abovementioned graphs are(12)HξSLn=∑gs∈E12dg+dsξ+∑gs∈E22dg+dsξ+∑gs∈E32dg+dsξ+∑gs∈E42dg+dsξ=24⋅5ξ+2ξ+1⋅3ξn−3+2⋅7ξn−2+23ξ+12n2−10n+12,GAξSLn=∑gs∈E12dg⋅dsξdg+ds+∑gs∈E22dg⋅dsξdg+ds+∑gs∈E32dg⋅dsξdg+ds+∑gs∈E42dg⋅dsξdg+ds=26ξ5E1+29ξ6E2+212ξ7E3+216ξ8E4=2ξ+4⋅3ξ5+22⋅32ξ−1n−3+22ξ+3⋅3ξn−27+24ξ−22n2−10n+12,KB1ξSLn=∑gs∈E1S(eg+Sesξ+∑gs∈E2S(eg+Sesξ+∑gs∈E3S(eg+Sesξ+∑gs∈E4S(eg+Sesξ=2ξ+3⋅32ξ+0⋅2352ξ+2ξ+2n−513ξ+23ξ+2ξ+3⋅17ξ+22ξ+2⋅32ξn−4+23⋅35ξ+24ξ⋅3ξ2n2−14n+24,KB2ξSLn=∑gs∈E1Seg⋅Sesξ+∑gs∈E2Seg⋅Sesξ+∑gs∈E3Seg⋅Sesξ+∑gs∈E4Seg⋅Sesξ=23ξ+3⋅32ξ+22ξ+3⋅39ξ+n−522⋅132ξ+23⋅232ξ+23ξ+3⋅33ξ+22⋅299ξn−4+2ξ+3⋅253ξ+26ξ⋅32ξ2n2−14n+24,YξSLn=∑gs∈E1dgdg+dsdsξ+∑gs∈E2dgdg+dsdsξ+∑gs∈E3dgdg+dsdsξ+∑gs∈E4dgdg+dsdsξ=23⋅31ξ+22ξ+2⋅32ξn−3+22⋅283ξn−2+29ξ2n2−10n+12.


## 6. Wheel GraphWn

Theorem 7.
LetWn be wheel graph having order n+1; then,(13)HξWn=2n2ξ⋅3ξ+3+nξ,GAξWn=32ξ−1n+2⋅3ξnξ+13+n−1,KB1ξWn=18+2nξ+n2+2n+9ξn,KB2ξWn=9+n2ξn+nξ+1n2+10n+9ξ,YξWn=2ξ⋅33ξ+27+nnn.Proof.
LetWn be wheel graph; then, n+1 and 2n are edges and vertices, respectively. For the desired topological indices, we made two partitions of the edge set that are E1 and E2. The partition E1 consists of n edges gs such that dg=3 and ds=3. The second partition E2 consists of n edges gs such that dg=3 and ds=n+1. The values for the stated indices are calculated in the following:(14)HξWn=∑gs∈E12dg+dsξ+∑gs∈E22dg+dsξ=2n2ξ⋅3ξ+3+nξ,GAξWn=∑gs∈E12dg⋅dsξdg+ds+∑gs∈E22dg⋅dsξdg+ds=32ξ−1n+2⋅3ξnξ+13+n−1,KB1ξWn=∑gs∈E1Seg+Sesξ+∑gs∈E2Seg+Sesξ=9+n+9+nξE1+9+n+nn+1ξE2=18+2nξ+n2+2n+9ξn,KB2ξWn=∑gs∈E1Seg⋅Sesξ+∑gs∈E2Seg⋅Sesξ=9+n⋅9+nξE1+9+n⋅nn+1ξE2=9+n2ξn+nξ+1n2+10n+9ξ,YξWn=∑gs∈E1dgdg+dsdsξ+∑gs∈E2dgdg+dsdsξ=2ξ⋅33ξ+27+nnn.


## 7. Gear Graph,GRn

Theorem 8.
LetGRn be gear graph; then,(15)HξGRn=22⋅5ξn+2n3+nξ,GAξGRn=2ξ+2⋅3ξ⋅5−1⋅n+2⋅3ξ⋅nξ+13+n−1,KB1ξGRn=2n13+nξ+nn2+2n+7ξ,KB2ξGRn=n2ξ+1⋅3ξ+nξn+1ξ7+nξ,YξGRn=2⋅31ξ+27+nnξn.Proof.
LetGRn be gear graph; then, 2n+1 and 3n are edges and vertices, respectively. For values of the desired topological indices, we divide the edge set into two partition E1 and E2. The partition E1 consists of 2n edges gs, such that dg=2 and ds=3. The partition E2 consists of n edges gs, such that dg=3 and ds=n. For the stated indices, we have the following:(16)HξGRn=∑gs∈E12dg+dsξ+∑gs∈E22dg+dsξ=22⋅5ξn+2n3+nξ,GAξGRn=∑gs∈E12dg⋅dsξdg+ds+∑gs∈E22dg⋅dsξdg+ds=2ξ+2⋅3ξ⋅5−1⋅n+2⋅3ξ⋅nξ+13+n−1,KB1ξGRn=∑gs∈E1Seg+Sesξ+∑gs∈E2Seg+Sesξ=2n13+nξ+nn2+2n+7ξ,KB2ξGRn=∑gs∈E1Seg⋅Sesξ+∑gs∈E2Seg⋅Sesξ=n2ξ+1⋅3ξ+nξn+1ξ7+nξ,YξGRn=∑gs∈E1dgg+dsdsξ+∑gs∈E2dgg+dsdsξ=2⋅31ξ+27+nnξ.


## 8. Helm GraphHn

Theorem 9.
LetHn be a helm graph; then,(17)HξHn=5ξ+23ξ+4+nξ2n,GAξHn=22ξ⋅5−1+24ξ−3+22ξ⋅nξ4+n−12n,KB1ξHn=20+nξ+2ξ17+nξ+n2+3n+17ξn,KB2ξHn=n17+nξ3ξ+17+nξ+nξ2+nξ,YξHn=257ξ+29ξ+256+nnξn.Proof..
LetHn be a helm graph; then, 2n+1 and 3n are vertices and edges, respectively. For values of the desired topological indices, we divide the edge set into three partitions say E1, E2, and E3. The partition E1 consists of n edges gs such that dg=1 and ds=4. The partition E2 consists of n edges gs such that dg=4 and ds=4. The partition E3 consists of n edges gs such that dg=4 and ds=n. The desired values about the stated topological indices are the following:(18)HξHn=∑gs∈E12dg+dsξ+∑gs∈E22dg+dsξ+∑gs∈E32dg+dsξ=5ξ+23ξ+4+nξ2n,GAξHn=∑gs∈E12dg⋅dsξdg+ds+∑gs∈E12dg⋅dsξdg+ds+∑gs∈E22dg⋅dsξdg+ds=22ξ⋅5−1+24ξ−3+22ξ⋅nξ4+n−12n,KB1ξHn=∑gs∈E1Seg+Sesξ+∑gs∈E2Seg+Sesξ+∑gs∈E3Seg+Sesξ=20+nξ+2ξ17+nξ+n2+3n+17ξn,KB2ξHn=∑gs∈E1Seg⋅Sesξ+∑gs∈E2Seg⋅Sesξ+∑gs∈E3Seg⋅Sesξ=n17+nξ3ξ+17+nξ+nξ2+nξ,YξHn=∑gs∈E1dgdg+dsdsξ+∑gs∈E2dgdg+dsdsξ+∑gs∈E3dgdg+dsdsξ.


## 9. Chemical Applicability of the Stated Indices in Generalized Form

In quantitative structure property relationship (QSPR) and quantitative structure activity relationship (QSAR), topological indices with higher correlation factor are of the most interest. In this section of the paper, we are going to discuss the linear regression analysis of the generalized form of the aforementioned topological indices with entropy(S), acentric factor (AcentFac), enthalpy of vaporization (HVAP), and standard enthalpy of vaporization (DHVAP) of octane isomers, which gives very closed relation regarding different properties of the stated chemicals which are labeled in Tables1–10, where some values are found by plugging different values of ξ in the generalized obtained results. We also presented fitted models for the residuals that are obtained for some of the stated indices. The generalized form of the stated indices is tested using a data of octane isomers that are found in [11]. Statistical calculation of entropy of dominating David derived networks can be found in [18], for further study regarding different properties and uses of indices in different chemicals also in other fields of science and technology can be found in the [11, 18–20] and the references therein. Using the generalized values from the above table, we get the following interesting approximations for octane isomers which is of the most interest. Figure 5 represents scatter diagram of S on Hξ superimposed by the fitted regression line. Figure 6 represents the residuals for S on Hξ. Figure 7 represents scatter plot of S on GAξ superimposed by the fitted regression line. Figure 8 represents the residuals for S on GAξ. Figure 9 represents scatter diagram of S on KB1ξ superimposed by the fitted regression line. Figure 10 is for the residuals for S on KB1ξ.Table 1
Experimental values of entropy(S) and the corresponding values of generalized form of harmonic index, geometric arithmetic index, and first Kulli–Basava index of octane isomers.
AlkaneSHξGAξKB1ξn-octane111.674.3ξ+10.4ξ2ξ+2/3+10.4ξ−12.4ξ+2.7ξ+3.8ξ2-methylheptane109.842.3ξ+10.4ξ+2.5ξ2ξ/3+6.4ξ−1+2.6ξ/5+3ξ3ξ+12ξ+2.14ξ+16ξ+20ξ+35ξ3-methylheptane111.2622.3ξ+6.4ξ+22.5ξ2ξ+2+4ξ+3ξ+2.6ξ/54ξ+7ξ+9ξ+13ξ+10ξ+12ξ+5ξ4-methylheptane109.3222.3ξ+6.4ξ+4.5ξ2ξ+2/3+4ξ+3ξ/4+4.6ξ2.4ξ+2.8ξ+10ξ+2.13ξ3-ethylhexane109.436.3ξ+6.5ξ+2.4ξ6.2ξ/3+6ξ+1/5+2⋅4ξ−12.5ξ+2.13ξ+14ξ+8ξ+4ξ2,2-dimethylhexane103.423.5ξ+6ξ+2.4ξ+3ξ6.4ξ/5+4ξ+8ξ/3+2ξ+1/33.1ξ+6ξ+19ξ+10ξ+7ξ+4ξ2,3-dimethylhexane108.0222ξ+2+5ξ+6ξ+3ξ6.3ξ/4+9ξ/3+2.4ξ−1+2.6ξ/5+2ξ+1/32.10ξ+17ξ+11ξ+13ξ+7ξ+4ξ2,4-dimethylhexane106.982.3ξ+6.4ξ+6.5ξ2ξ+1/3+6.3ξ/4+6.6ξ/55ξ+2.9ξ+10ξ+14ξ+13ξ+12ξ2,5-dimethylhexane105.7223.4ξ+4.5ξ+2.4ξ2.3ξ+4.6ξ/5+2.4ξ−14.9ξ+2.12ξ+10ξ3,3-dimethylhexane104.744.3ξ+4.6ξ+4.5ξ+2.4ξ4.2ξ/3+4.8ξ/6+4ξ+1/5+2⋅4ξ−14ξ+6ξ+9ξ+2.17ξ+19ξ+20ξ3,4-dimethylhexane106.594.3ξ+4.5ξ+4.4ξ+2.6ξ2ξ+2/3+4⋅6ξ/5+3ξ+9ξ/62.5ξ+2.11ξ+2.13ξ+18ξ2methyl-3ethylpentane106.064.3ξ+4.5ξ+2.6ξ+4ξ+12ξ+2/3+4.6ξ/5+2.9ξ/6+4.3ξ/42.5ξ+2.14ξ+2.10ξ+18ξ3methyl-3ethylpentane101.486.3ξ+6ξ+1+2.5ξ2ξ+1+8ξ+22ξ+1/53.6ξ3.20ξ+18ξ2,2,3-trimethyl-pentane101.316.5ξ+2.7ξ+2.4ξ+2.5ξ+2.3ξ6.4ξ/5+2.12ξ/7+2.3ξ/4+2.66/5+2ξ+1/33.16ξ+23ξ+16ξ+14ξ+5ξ2,2,4-trimethyl-pentane104.096.5ξ+2.6ξ+2.5ξ+4ξ+16.4ξ/5+8ξ/3+2.6ξ/5+3ξ3.16ξ+20ξ+14ξ+2.9ξ2,3,3-trimethyl-pentane102.062.3ξ+2.6ξ+4.5ξ+2.7ξ+4ξ+12ξ+1/3+8ξ/3+4ξ+1/5+2.12ξ/7+3ξ6ξ+20ξ+2.18ξ+24ξ+2.11ξ2,3,4-trimethyl-pentane102.3910.4ξ+4.6ξ10.3ξ/4+4.9ξ/64.10ξ+2.18ξ+12ξ2,2,3,3-tetramethyl-butane93.0612.5ξ+2.8ξ12.4ξ/5+2.16ξ/86.18ξ+30ξTable 2
Experimental values of entropy(S) and the corresponding values of generalized form of harmonic index, geometric arithmetic index, and first Kulli–Basava index of octane isomers for different values ofξ.
AlkaneSHξGAξKB1ξn-octane111.67111.483 ifξ=1.576111.665 ifξ=2.683111.672 ifξ=1.45912-methylheptane109.84109.843 ifξ=1.417109.845 ifξ=2.5587109.882 ifξ=0.98753-methylheptane111.26111.27 ifξ=1.486111.27 ifξ=2.54111.26 ifξ=1.27614-methylheptane109.32109.389 ifξ=1.474109.331 ifξ=1.749109.324 ifξ=1.26623-ethylhexane109.43109.434 ifξ=1.47109.43 ifξ=2.3807109.437 ifξ=1.24582,2-dimethylhexane103.42103.431 ifξ=1.755103.426 ifξ=2.3423103.354 ifξ=1.06112,3-dimethylhexane108.02108.146 ifξ=1.676108.032 ifξ=2.1141108.024 ifξ=1.1682,4-dimethylhexane106.98106.934 ifξ=1.392106.954 ifξ=2.3666106.986 ifξ=1.16662,5-dimethylhexane105.72105.685 ifξ=1.387105.783 ifξ=2.431105.599 ifξ=1.17783,3-dimethylhexane104.74104.749ξ=1.315104.369 ifξ=2.204.745 ifξ=1.0483,4-dimethylhexane106.59106.527 ifξ=1.384106.581 ifξ=2.3876106.583 ifξ=1.13652methyl-3ethylpentane106.06106.063 ifξ=1.3811106.064 ifξ=2.246106.055 ifξ=1.13413methyl-3ethylpentane101.48101.408 ifξ=1.2911101.437 ifξ=2.136101.405 ifξ=1.01992,2,3-trimethyl-pentane101.31101.303 ifξ=1.247101.362 ifξ=1.8765101.392 ifξ=0.9842,2,4-trimethyl-pentane104.09104.002 ifξ=1.267104.084 ifξ=2.2889104.098 if ξ=1.01492,3,3-trimethyl-pentane102.06102.067 ifξ=1.2505102.333 ifξ=2.08102.067 ifξ=0.97992,3,4-trimethyl-pentane102.39102.391 ifξ=1.3043102.391 ifξ=2.154102.172 ifξ=1.05812,2,3,3-tetramethyl-butane93.0693.062 ifξ=1.118493.0975 ifξ=2.0193.1717 ifξ=0.869Table 3
Experimental values of acentric factor and the corresponding values of generalized form of harmonic index, geometric arithmetic index, and first Kulli–Basava index of octane isomers.
AlkaneAcentFacHξGAξKB1ξn-octane0.3978984.3ξ+10.4ξ2ξ+2/3+10.4ξ−12.4ξ+2.7ξ+3.8ξ2-methylheptane0.3779162.3ξ+10.4ξ+2.5ξ2ξ/3+6.4ξ−1+2⋅6ξ/5+3ξ3ξ+12ξ+2.14ξ+16ξ+20ξ+35ξ3-methylheptane0.37100222.3ξ+6.4ξ+22+5ξ2ξ+2+4ξ+3ξ+2⋅6ξ/54ξ+7ξ+9ξ+13ξ+10ξ+12ξ+5ξ4-methylheptane0.37150422.3ξ+6.4ξ+4.5ξ2ξ+2/3+4ξ+3ξ/4+4.6ξ2.4ξ+2.8ξ+10ξ+2.13ξ3-ethylhexane0.3624726.3ξ+6.5ξ+2.4ξ6⋅2ξ/3+6ξ+1/5+2.4ξ−12.5ξ+2.13ξ+14ξ+8ξ+4ξ2,2-dimethylhexane0.3394263.5ξ+6ξ+2.4ξ+3ξ6⋅4ξ/5+4ξ+8ξ/3+2ξ+1/33.16ξ+19ξ+10ξ+7ξ+4ξ2,3-dimethylhexane0.34824722ξ+2+5ξ+6ξ+3ξ6.3ξ/4+9ξ/3+2.4ξ−1+2.6ξ/5+2ξ+1/32.10ξ+17ξ+11ξ+13ξ+7ξ+4ξ2,4-dimethylhexane0.3442232.3ξ+6.4ξ+6.5ξ2ξ+1/3+6⋅3ξ/4+6⋅6ξ/55ξ+2.9ξ+10ξ+14ξ+13ξ+12ξ2,5-dimethylhexane0.3568323.4ξ+4.5ξ+2.4ξ2.3ξ+4.6ξ/5+2.4ξ−14.9ξ+2.12ξ+10ξ3,3-dimethylhexane0.3225964.3ξ+4.6ξ+4.5ξ+2.4ξ4.2ξ/3+4.8ξ/6+4ξ+1/5+2.4ξ−14ξ+6ξ+9ξ+2.17ξ+19ξ+20ξ3,4-dimethylhexane0.3403454.3ξ+4.5ξ+4.4ξ+2.6ξ2ξ+2/3+4.6ξ/5+3ξ+9ξ/62.5ξ+2.11ξ+2.13ξ+18ξ2methyl-3ethylpentane0.3324334.3ξ+4.5ξ+2.6ξ+4ξ+12ξ+2/3+4.6ξ/5+2.9ξ/6+4.3ξ/42.5ξ+2.14ξ+2.10ξ+18ξ3methyl-3ethylpentane0.3068996.3ξ+6ξ+1+2.5ξ2ξ+1+8ξ+22ξ+1/53.6ξ3.20ξ+18ξ2,2,3-trimethyl-pentane0.3008166.5ξ+2.7ξ+2.4ξ+2.5ξ+2.3ξ6.4ξ/5+2.12ξ/7+2.3ξ/4+2.66/5+2ξ+1/33.16ξ+23ξ+16ξ+14ξ+5ξ2,2,4-trimethyl-pentane0.305376.5ξ+2.6ξ+2.5ξ+4ξ+16.4ξ/5+8ξ/3+2.6ξ/5+3ξ3.16ξ+20ξ+14ξ+2.9ξ2,3,3-trimethyl-pentane0.2931772.3ξ+2.6ξ+4.5ξ+2.7ξ+4ξ+12ξ+1/3+8ξ/3+4ξ+1/5+2.12ξ/7+3ξ6ξ+20ξ+2.18ξ+24ξ+2.11ξ2,3,4-trimethyl-pentane0.31742210.4ξ+4.6ξ10.3ξ/4+4.9ξ/64.10ξ+2.18ξ+12ξ2,2,3,3-tetramethyl-butane0.25529412.5ξ+2.8ξ12.4ξ/5+2.16ξ/86.18ξ+30ξTable 4
Experimental values of acentric factor and the corresponding values of generalized form of harmonic index, geometric arithmetic index, and first Kulli–Basava index of octane isomers for different values ofξ.
AlkaneAcentFacHξGAξKB1ξn-octane0.3978980.390726 ifξ=−2.80.397095 ifξ=−2.2320.393803 ifξ=−1.632-methylheptane0.3779160.377722 ifξ=−2.870.373743 ifξ=−1.80.37766 ifξ=−1.3353-methylheptane0.3710020.370688 ifξ=−2.760.377933 ifξ=−3.520.375643 ifξ=−1.514-methylheptane0.3715040.370266 ifξ=−2.76090.374663 ifξ=−2.360.371555 ifξ=−1.5613-ethylhexane0.3624720.372775 ifξ=−2.8250.36575 ifξ=−2.560.362373 ifξ=−1.5712,2-dimethylhexane0.3394260.333752 ifξ=−2.10.339447 ifξ=−1.910.333496 ifξ=−1.392,3-dimethylhexane0.3482470.341947 ifξ=−4.610.345979 ifξ=−2.3010.342585 ifξ=−1.442,4-dimethylhexane0.3442230.340783 ifξ=−2.660.342507 ifξ=−2.070.344292 ifξ=−1.3662,5-dimethylhexane0.356830.342116 ifξ=−2.5810.358332 ifξ=−1.780.356664 ifξ=−1.3033,3-dimethylhexane0.3225960.337126 ifξ=−2.69950.324943 ifξ=−3.120.322704 ifξ=−1.4283,4-dimethylhexane0.3403450.345705 if ξ=−2.750.340694 ifξ=−1.8720.340781 ifξ=−1.4172methyl-3ethylpentane0.3324330.332769 ifξ=−2.780.33118 ifξ=−1.910.336483 ifξ=−1.433methyl-3ethylpentane0.3068990.307387 ifξ=−2.880.302726 ifξ=−2.78010.303201 ifξ=−1.4062,2,3-trimethyl-pentane0.3008160.30815 ifξ=−2.590.300499 ifξ=−2.50.300757 ifξ=−1.2712,2,4-trimethyl-pentane0.305370.305828 ifξ=−2.4670.307001 ifξ=−1.6740.304683 ifξ=−1.22012,3,3-trimethyl-pentane0.2931770.291389 ifξ−2.660.293276 ifξ=−2.1390.295659 ifξ=−1.262,3,4-trimethyl-pentane0.3174220.318999 ifξ=−2.580.317756 ifξ=−1.9070.315898 ifξ=−1.262,2,3,3-tetramethyl-butane0.2552940.255116 ifξ=−2.4250.255169 ifξ=−2.2450.255542 ifξ=−1.123Table 5
Experimental values of DHVAP and the corresponding values of generalized form of harmonic index, geometric arithmetic index, and first Kulli–Basava index of octane isomers.
AlkaneDHVAPHξGAξKB1ξn-octane9.9154.3ξ+10.4ξ2ξ+2/3+10.4ξ−12.4ξ+2.7ξ+3.8ξ2-Methylheptane9.4842.3ξ+10.4ξ+2.5ξ2ξ/3+6.4ξ−1+2.6ξ/5+3ξ3ξ+12ξ+2.14ξ+16ξ+20ξ+35ξ3-Methylheptane9.52122.3ξ+6.4ξ+22+5ξ2ξ+2+4ξ+3ξ+2⋅6ξ/54ξ+7ξ+9ξ+13ξ+10ξ+12ξ+5ξ4-Methylheptane9.48322.3ξ+6.4ξ+4.5ξ2ξ+2/3+4ξ+3ξ/4+4.6ξ2.4ξ+2.8ξ+10ξ+2.13ξ3-Ethylhexane9.4766.3ξ+6.5ξ+2.4ξ6.2ξ/3+6ξ+1/5+2.4ξ−12.5ξ+2.13ξ+14ξ+8ξ+4ξ2,2-dimethylhexane8.9153.5ξ+6ξ+2.4ξ+3ξ6.4ξ/5+4ξ+8ξ/3+2ξ+1/33.16ξ+19ξ+10ξ+7ξ+4ξ2,3-dimethylhexane9.27222ξ+2+5ξ+6ξ+3ξ6.3ξ/4+9ξ/3+2.4ξ−1+2.6ξ/5+2ξ+1/32.10ξ+17ξ+11ξ+13ξ+7ξ+4ξ2,4-dimethylhexane9.0292.3ξ+6.4ξ+6.5ξ2ξ+1/3+6.3ξ/4+6.6ξ/55ξ+2.9ξ+10ξ+14ξ+13ξ+12ξ2,5-dimethylhexane9.05123.4ξ+4.5ξ+2.4ξ2.3ξ+4.6ξ/5+2.4ξ−14.9ξ+2.12ξ+10ξ3,3-dimethylhexane8.9734.3ξ+4.6ξ+4.5ξ+2.4ξ4.2ξ/3+4.8ξ/6+4ξ+1/5+2.4ξ−14ξ+6ξ+9ξ+2.17ξ+19ξ+20ξ3,4-dimethylhexane9.3164.3ξ+4.5ξ+4.4ξ+2.6ξ2ξ+2/3+4.6ξ/5+3ξ+9ξ/62.5ξ+2.11ξ+2.13ξ+18ξ2methyl-3ethylpentane9.2094.3ξ+4.5ξ+2.6ξ+4ξ+12ξ+2/3+4.6ξ/5+2.9ξ/6+4.3ξ/42.5ξ+2.14ξ+2.10ξ+18ξ3methyl-3ethylpentane9.0816.3ξ+6ξ+1+2.5ξ2ξ+1+8ξ+22ξ+1/53.6ξ3.20ξ+18ξ2,2,3-trimethyl-pentane8.8266.5ξ+2.7ξ+2.4ξ+2.5ξ+2.3ξ6.4ξ/5+2.12ξ/7+2.3ξ/4+2.66/5+2ξ+1/33.16ξ+23ξ+16ξ+14ξ+5ξ2,2,4-trimethyl-pentane8.4026.5ξ+2.6ξ+2.5ξ+4ξ+16.4ξ/5+8ξ/3+2.6ξ/5+3ξ3.16ξ+20ξ+14ξ+2.9ξ2,3,3-trimethyl-pentane8.8972.3ξ+2.6ξ+4.5ξ+2.7ξ+4ξ+12ξ+1/3+8ξ/3+4ξ+1/5+2.12ξ/7+3ξ6ξ+20ξ+2.18ξ+24ξ+2.11ξ2,3,4-trimethyl-pentane9.01410.4ξ+4.6ξ10.3ξ/4+4.9ξ/64.10ξ+2.18ξ+12ξ2,2,3,3-tetramethyl-butane8.4112.5ξ+2.8ξ12.4ξ/5+2.16ξ/86.18ξ+30ξTable 6
Experimental values of HVAP and the corresponding values of generalized form of harmonic index, geometric arithmetic index, and first Kulli–Basava index of octane isomers.
AlkaneHVAPHξGAξKB1ξn-octane73.194.3ξ+10.4ξ2ξ+2/3+10.4ξ−12.4ξ+2.7ξ+3.8ξ2-methylheptane70.32.3ξ+10.4ξ+2.5ξ2ξ/3+6.4ξ−1+2⋅6ξ/5+3ξ3ξ+12ξ+2.14ξ+16ξ+20ξ+35ξ3-methylheptane71.322.3ξ+6.4ξ+22+5ξ2ξ+2+4ξ+3ξ+2⋅6ξ/54ξ+7ξ+9ξ+13ξ+10ξ+12ξ+5ξ4-methylheptane70.9122.3ξ+6.4ξ+4.5ξ2ξ+2/3+4ξ+3ξ/4+4.6ξ2.4ξ+2.8ξ+10ξ+2.13ξ3-ethylhexane71.76.3ξ+6.5ξ+2.4ξ6.2ξ/3+6ξ+1/5+2.4ξ−12.5ξ+2.13ξ+14ξ+8ξ+4ξ2,2-dimethylhexane67.73.5ξ+6ξ+2.4ξ+3ξ6.4ξ/5+4ξ+8ξ/3+2ξ+1/33.16ξ+19ξ+10ξ+7ξ+4ξ2,3-dimethylhexane70.222ξ+2+5ξ+6ξ+3ξ6.3ξ/4+9ξ/3+2.4ξ−1+2.6ξ/5+2ξ+1/32.10ξ+17ξ+11ξ+13ξ+7ξ+4ξ2,4-dimethylhexane68.52.3ξ+6.4ξ+6.5ξ2ξ+1/3+6.3ξ/4+6.6ξ/55ξ+2.9ξ+10ξ+14ξ+13ξ+12ξ2,5-dimethylhexane68.623.4ξ+4.5ξ+2.4ξ2.3ξ+4.6ξ/5+2.4ξ−14.9ξ+2.12ξ+10ξ3,3-dimethylhexane68.54.3ξ+4.6ξ+4.5ξ+2.4ξ4.2ξ/3+4.8ξ/6+4ξ+1/5+2.4ξ−14ξ+6ξ+9ξ+2.17ξ+19ξ+20ξ3,4-dimethylhexane70.24.3ξ+4.5ξ+4.4ξ+2.6ξ2ξ+2/3+4.6ξ/5+3ξ+9ξ/62.5ξ+2.11ξ+2.13ξ+18ξ2methyl-3ethylpentane69.74.3ξ+4.5ξ+2.6ξ+4ξ+12ξ+2/3+4.6ξ/5+2.9ξ/6+4.3ξ/42.5ξ+2.14ξ+2.10ξ+18ξ3methyl-3ethylpentane69.36.3ξ+6ξ+1+2.5ξ2ξ+1+8ξ+22ξ+1/53.6ξ3.20ξ+18ξ2,2,3-trimethyl-pentane67.36.5ξ+2.7ξ+2.4ξ+2.5ξ+2.3ξ6.4ξ/5+2.12ξ/7+2.3ξ/4+2.66/5+2ξ+1/33.16ξ+23ξ+16ξ+14ξ+5ξ2,2,4-trimethyl-pentane64.876.5ξ+2.6ξ+2.5ξ+4ξ+16.4ξ/5+8ξ/3+2.6ξ/5+3ξ3.16ξ+20ξ+14ξ+2.9ξ2,3,3-trimethyl-pentane68.12.3ξ+2.6ξ+4.5ξ+2.7ξ+4ξ+12ξ+1/3+8ξ/3+4ξ+1/5+2.12ξ/7+3ξ6ξ+20ξ+2.18ξ+24ξ+2.11ξ2,3,4-trimethyl-pentane68.3710.4ξ+4.6ξ10.3ξ/4+4.9ξ/64.10ξ+2.18ξ+12ξ2,2,3,3-tetramethyl-butane66.212.5ξ+2.8ξ12.4ξ/5+2.16ξ/86.18ξ+30ξTable 7
Experimental values of entropy and the corresponding values of generalized form of second Kulli–Basava index and generalized power-sum-connectivity index of octane isomers.
AlkaneSKB2ξYξn-octane111.672.3ξ+2.12ξ+3.16ξ2.5ξ+5.8ξ2-methylheptane109.843ξ+12ξ+2.14ξ+16ξ+20ξ+35ξ4ξ+3.8ξ+31ξ+2.28ξ3-methylheptane111.263ξ+12ξ+20ξ+40ξ+16ξ+32ξ+4ξ2.5ξ+2.8ξ+2.31ξ+28ξ4-methylheptane109.322.3ξ+2.15ξ+16ξ+2.40ξ2.5ξ+2.8ξ+31ξ+28ξ3-ethylhexane109.432.4ξ+2.36ξ+45ξ+15ξ+3ξ3.5ξ+3.31ξ+8ξ2,2-dimethylhexane103.423.39ξ+78ξ+24ξ+12ξ+3ξ3.257ξ+2.8ξ+260ξ+5ξ2,3-dimethylhexane108.022.16ξ+72ξ+18ξ+36ξ+12ξ+4ξ3.28ξ+54ξ+31ξ+8ξ+5ξ2,4-dimethylhexane106.984ξ+16ξ+32ξ+48ξ+42ξ+2.14ξ5ξ3.28ξ+3.31ξ2,5-dimethylhexane105.724.14ξ+2.35ξ+25ξ4.28ξ+2.28ξ+8ξ3,3-dimethylhexane104.743ξ+5ξ+18ξ+2.52ξ+70ξ+84ξ2.5ξ+2.260ξ+2.257ξ+8ξ3,4-dimethylhexane106.592.4ξ+2.18ξ+2.36ξ+81ξ2.5ξ+2.31ξ+2.28ξ+54ξ2methyl-3ethylpentane106.062.4ξ+2.40ξ+2.16ξ+80ξ2.5ξ+2.31ξ+54ξ+2.28ξ3methyl-3ethylpentane101.483.5ξ+3.75ξ+45ξ3.5ξ+3.260ξ+257ξ2,2,3-trimethyl-pentane101.313.39ξ+130ξ+60ξ+40ξ+4ξ3.257ξ+283ξ+28ξ+31ξ+5ξ2,2,4-trimethyl-pentane104.093.39ξ+91ξ+49ξ+2.14ξ3.257ξ+260ξ+31ξ+2.28ξ2,3,3-trimethyl-pentane102.065ξ+75ξ+2.45ξ+135ξ+2.18ξ5ξ+260ξ+2.257ξ+283ξ+2.28ξ2,3,4-trimethyl-pentane102.394.16ξ+2.80ξ+20ξ5.28ξ+2.54ξ2,2,3,3-tetramethyl-butane93.066.45ξ+225ξ6.257ξ+512ξTable 8
Experimental values of acentric factor and the corresponding values of generalized form of second Kulli–Basava index and generalized power-sum-connectivity index of octane isomers.
AlkaneAcentFacKB2ξYξn-octane0.3978982.3ξ+2.12ξ+3.16ξ2.5ξ+5.8ξ2-methylheptane0.3779163ξ+12ξ+2.14ξ+16ξ+20ξ+35ξ4ξ+3.8ξ+31ξ+2.28ξ3-methylheptane0.3710023ξ+12ξ+20ξ+40ξ+16ξ+32ξ+4ξ2.5ξ+2.8ξ+2.31ξ+28ξ4-methylheptane0.3715042.3ξ+2.15ξ+16ξ+2.40ξ2.5ξ+2.8ξ+31ξ+28ξ3-ethylhexane0.3524722.4ξ+2.36ξ+45ξ+15ξ+3ξ3.5ξ+3.31ξ+8ξ2,2-dimethylhexane0.3394263.39ξ+78ξ+24ξ+12ξ+3ξ3.257ξ+2.8ξ+260ξ+5ξ2,3-dimethylhexane0.3482472.16ξ+72ξ+18ξ+36ξ+12ξ+4ξ3.28ξ+54ξ+31ξ+8ξ+5ξ2,4-dimethylhexane0.3442234ξ+16ξ+32ξ+48ξ+42ξ+2.14ξ5ξ3.28ξ+3.31ξ2,5-dimethylhexane0.356834.14ξ+2.35ξ+25ξ4.28ξ+2.28ξ+8ξ3,3-dimethylhexane0.3225963ξ+5ξ+18ξ+2.52ξ+70ξ+84ξ2.5ξ+2.260ξ+2.257ξ+8ξ3,4-dimethylhexane0.3403452.4ξ+2.18ξ+2.36ξ+81ξ2.5ξ+2.31ξ+2.28ξ+54ξ2methyl-3ethylpentane0.3324332.4ξ+2.40ξ+2.16ξ+80ξ2.5ξ+2.31ξ+54ξ+2.28ξ3methyl-3ethylpentane0.3068993.5ξ+3.75ξ+45ξ3.5ξ+3.260ξ+257ξ2,2,3-trimethyl-pentane0.3008163.39ξ+130ξ+60ξ+40ξ+4ξ3.257ξ+283ξ+28ξ+31ξ+5ξ2,2,4-trimethyl-pentane0.305373.39ξ+91ξ+49ξ+2.14ξ3.257ξ+260ξ+31ξ+2.28ξ2,3,3-trimethyl-pentane0.2931775ξ+75ξ+2.45ξ+135ξ+2.18ξ5ξ+260ξ+2.257ξ+283ξ+2.28ξ2,3,4-trimethyl-pentane0.3174224.16ξ+2.80ξ+20ξ5.28ξ+2.54ξ2,2,3,3-tetramethyl-butane0.2552946.45ξ+225ξ6.257ξ+512ξTable 9
Experimental values of DHVAP and the corresponding values of generalized form of second Kulli–Basava index and generalized power-sum-connectivity index of octane isomers.
AlkaneDHVAPKB2ξYξn-octane9.9152.3ξ+2.12ξ+3.16ξ2.5ξ+5.8ξ2-methylheptane9.4843ξ+12ξ+2.14ξ+16ξ+20ξ+35ξ4ξ+3.8ξ+31ξ+2.28ξ3-methylheptane9.5213ξ+12ξ+20ξ+40ξ+16ξ+32ξ+4ξ2.5ξ+2.8ξ+2.31ξ+28ξ4-methylheptane9.4832.3ξ+2.15ξ+16ξ+2.40ξ2.5ξ+2.8ξ+31ξ+28ξ3-ethylhexane9.4762.4ξ+2.36ξ+45ξ+15ξ+3ξ3.5ξ+3.31ξ+8ξ2,2-dimethylhexane8.9153.39ξ+78ξ+24ξ+12ξ+3ξ3.257ξ+2.8ξ+260ξ+5ξ2,3-dimethylhexane9.2722.16ξ+72ξ+18ξ+36ξ+12ξ+4ξ3.28ξ+54ξ+31ξ+8ξ+5ξ2,4-dimethylhexane9.0294ξ+16ξ+32ξ+48ξ+42ξ+2.14ξ5ξ3.28ξ+3.31ξ2,5-dimethylhexane9.0514.14ξ+2.35ξ+25ξ4.28ξ+2.28ξ+8ξ3,3-dimethylhexane8.9733ξ+5ξ+18ξ+2.52ξ+70ξ+84ξ2.5ξ+2.260ξ+2.257ξ+8ξ3,4-dimethylhexane9.3162.4ξ+2.18ξ+2.36ξ+81ξ2.5ξ+2.31ξ+2.28ξ+54ξ2methyl-3ethylpentane9.2092.4ξ+2.40ξ+2.16ξ+80ξ2.5ξ+2.31ξ+54ξ+2.28ξ3methyl-3ethylpentane9.0813.5ξ+3.75ξ+45ξ3.5ξ+3.260ξ+257ξ2,2,3-trimethyl-pentane8.8263.39ξ+130ξ+60ξ+40ξ+4ξ3.257ξ+283ξ+28ξ+31ξ+5ξ2,2,4-trimethyl-pentane8.4023.39ξ+91ξ+49ξ+2.14ξ3.257ξ+260ξ+31ξ+2.28ξ2,3,3-trimethyl-pentane8.8975ξ+75ξ+2.45ξ+135ξ+2.18ξ5ξ+260ξ+2.257ξ+283ξ+2.28ξ2,3,4-trimethyl-pentane9.0144.16ξ+2.80ξ+20ξ5.28ξ+2.54ξ2,2,3,3-tetramethyl-butane8.416.45ξ+225ξ6.257ξ+512ξTable 10
Experimental values of HVAP and the corresponding values of generalized form of second Kulli–Basava index and generalized power-sum-connectivity index of octane isomers.
AlkaneHVAPKB2ξYξn-octane73.192.3ξ+2.12ξ+3.16ξ2.5ξ+5.8ξ2-methylheptane70.33ξ+12ξ+2.14ξ+16ξ+20ξ+35ξ4ξ+3.8ξ+31ξ+2.28ξ3-methylheptane71.33ξ+12ξ+20ξ+40ξ+16ξ+32ξ+4ξ2.5ξ+2.8ξ+2.31ξ+28ξ4-methylheptane70.912.3ξ+2.15ξ+16ξ+2.40ξ2.5ξ+2.8ξ+31ξ+28ξ3-ethylhexane71.72.4ξ+2.36ξ+45ξ+15ξ+3ξ3.5ξ+3.31ξ+8ξ2,2-dimethylhexane67.73.39ξ+78ξ+24ξ+12ξ+3ξ3.257ξ+2.8ξ+260ξ+5ξ2,3-dimethylhexane70.22.16ξ+72ξ+18ξ+36ξ+12ξ+4ξ3.28ξ+54ξ+31ξ+8ξ+5ξ2,4-dimethylhexane68.54ξ+16ξ+32ξ+48ξ+42ξ+2.14ξ5ξ3.28ξ+3.31ξ2,5-dimethylhexane68.64.14ξ+2.35ξ+25ξ4.28ξ+2.28ξ+8ξ3,3-dimethylhexane68.53ξ+5ξ+18ξ+2.52ξ+70ξ+84ξ2.5ξ+2.260ξ+2.257ξ+8ξ3,4-dimethylhexane70.22.4ξ+2.18ξ+2.36ξ+81ξ2.5ξ+2.31ξ+2.28ξ+54ξ2methyl-3ethylpentane69.72.4ξ+2.40ξ+2.16ξ+80ξ2.5ξ+2.31ξ+54ξ+2.28ξ3methyl-3ethylpentane69.33.5ξ+3.75ξ+45ξ3.5ξ+3.260ξ+257ξ2,2,3-trimethyl-pentane67.33.39ξ+130ξ+60ξ+40ξ+4ξ3.257ξ+283ξ+28ξ+31ξ+5ξ2,2,4-trimethyl-pentane64.873.39ξ+91ξ+49ξ+2.14ξ3.257ξ+260ξ+31ξ+2.28ξ2,3,3-trimethyl-pentane68.15ξ+75ξ+2.45ξ+135ξ+2.18ξ5ξ+260ξ+2.257ξ+283ξ+2.28ξ2,3,4-trimethyl-pentane68.374.16ξ+2.80ξ+20ξ5.28ξ+2.54ξ2,2,3,3-tetramethyl-butane66.26.45ξ+225ξ6.257ξ+512ξFigure 5
Scatter diagram ofS on Hξ superimposed by the fitted regression line.Figure 6
Plot shows the residuals forS on Hξ.Figure 7
Scatter diagram ofS on GAξ superimposed by the fitted regression line.Figure 8
Plot shows the residuals forS on GAξ.Figure 9
Scatter diagram ofS on KB1ξ superimposed by the fitted regression line.Figure 10
Plot shows the residuals forS on KB1ξ.Figure11 represents scatter diagram of AcentFac on Hξ superimposed by the fitted regression line. Figure 12 is for the residuals regarding AcentFac on Hξ. Figure 13 represents scatter diagram of AcentFac on GAξ super imposed by the fitted regression line. Figure 14 is for the residuals for AcentFac on GAξ. Figure 15 represents scatter diagram of AcentFac on KB1ξ superimposed by the fitted regression line. Figure 16 represents the residuals of AcentFac on KB1ξ.Figure 11
Scatter diagram ofAcentFac on Hξ superimposed by the fitted regression line.Figure 12
Plot shows the residuals forAcentFac on Hξ.Figure 13
Scatter diagram ofAcentFac on GAξ superimposed by the fitted regression line.Figure 14
Plot shows the residuals forAcentFac on GAξ.Figure 15
Scatter diagram ofAcentFac on KB1ξ superimposed by the fitted regression line.Figure 16
Plot for the residuals ofAcentFac on KB1ξ.As for different values ofξ, we obtained interesting results regarding different properties of alkanes. The results obtained by means of this generalized forms are closed to that of experimentally obtained values. In the remaining table, we can also find the approximate values of AcentFac, DHVAP, and HVAP. We can find closed values for the desired properties of the stated chemical by assigning different values to ξ. We found some fitted models be using Mathematica for S that are(19)S=0.074493+0.99916Hξ,S=48.5838+0.537382GAξ,S=10.6755+0.896433KB1ξ.For AcentFac, we have the following fitted models:(20)AcentFac=−0.434412+0.00730372Hξ,AcentFac=0.521876+0.00810559GA1ξ,AcentFac=−0.44993+0.0074482KB1ξ.For the remaining properties, we can find the fitted models as for some properties are given above in the last two equations.


## 10. Conclusion

In this study, various degree-based topological indices are studied in their generalized form for various well-known and special graphs that are bridge graph over path, bridge graph over cycle, bridge graph over complete graph, square lattice graph, wheel graph, gear graph, and helm graph. There are various significant applications of these indices in graph theory and chemistry which can be found in literature. We also studied the generalized form of the stated indices for various properties of alkanes which provided good result to the experimental values. In future, one can study this generalized form for other aspects of various chemicals.



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*Source: 1369490-2022-04-26.xml*