# Computation of Polynomial Degree-Based Topological Descriptors of Indu-Bala Product of Two Paths

Authors: Ghazanfar Abbas; Muhammad Ibrahim
Journal: Journal of Chemistry
 (2021)

---

## Abstract

Cheminformatics is entirely a newly coined term that encompasses a field that includes engineering computer sciences along with basic sciences. As we all know, vertices and edges form a network whereas vertex and its degrees contribute to joining edges. The degree of vertex is very much dependent on a reasonable proportion of network properties. There is no doubt that a network has to have a reliance of different kinds of hub buses, serials, and other connecting points to constitute a system that is the backbone of cheminformatics. The Indu-Bala product of two graphsG1 and G2 has a special notation as described in Section 2. The attainment of this product is very much due to related vertices at to different places of G1∨G2. This study states we have found M-polynomial and degree-based topological indices for Indu-Bala product of two paths Pk and Pj for j,k≥2. We also give some graphical representation of these indices and analyzed them graphically.


---

## Body

## 1. Introduction

LetG=VG,EG be a simple and finite graph of order n. We denote the nonempty vertex set by VG and edge set by EG. The fields of chemistry information sciences and mathematics have undoubtedly revolutionized by cheminformatics. It is a new subject that is very much helpful in keeping the data and getting information about chemicals. For this purpose, i.e., keeping the data and storing information, a significant help can be taken from the theory represented by graph in order to make index factors. The study of molecules according to their structures and their different functions based on QSAR models is also called a biological activity. The indictors that represent topology are also known as a subsidiary of the biological activity. Topological indices can be calculated using simply points (atoms) and linkages (chemical bonds) in a graphical representation. A polynomial, numeric number, a sequence of numbers, or an array representing the full graph can be used to identify it, and these representations are meant to be calculated particularly for that graph. The values in mathematics serve as indicators that have a logical connection to the graph and its topology. These are the indicators that give various dimensions and kinds to topological indices from distance based to degree based, counting conjugal polynomials and graphs. In chemistry and especially in graph theory, the degree-based topological indices play an essential role. Precisely, we can say that Σ gives new shape to the index connected with topology from real numbers to its zenith. Various indicator networks are always present in an entangled form of links nodes and hubs in a network. For example, various networks have similarities in atomic structure or molecular structure, such as honeycomb, grid networks, and hexagonal. Topological properties of these networks are very interesting, which are studied in various aspects, such as minimum metric dimension of a honeycomb network in [1] and silicate network in [2], topological properties of this network in [3], and topological indicators of honeycomb, silicate, and hexagonal networks in [4]. As we study the evolution of the things biologically, different kinds of structures having six dimensions and beehive shapes come into our contact. Many authors have researched on this topic; Hayat et al. computed topological indices of some networks in [5] and for some interconnection networks in [6]. On organized populations, Perc et al. studied the evolutionary dynamics of group interactions in [7] and on coevolutionary games in [8], and Szolnoki et al. further worked on the impact of noise on cooperation in spatial public goods games in topology-independent ways in [9] and on importance of percolation for evolution of cooperation in [10]. Mathematical references have also been found in research of paraffin, Wiener’s approach [11]. Wiener invented the index that is also known as the route number. This topological descriptor formed the basis for the topological indices, in terms of theory and application in [12, 13]. Therefore, the topological indices in the chemical and quantitative literature are Weiner in [14], Zagreb in [15], and Randic in [16]. The Indu-Bala product G1▼G2 of graphs G1 and G2 is obtained from two disjoint copies of the join G1∨G2 of G1 and G2 by joining the corresponding vertices in the two copies of G2.In this paper, we calculated some well-known topological indicators based on M-polynomial and degree based indices for Indu-Bala product of two paths.(1)MG,z1,z2=∑θ≤i≤j≤ϑmijGz1iz2j.As in [17], θ=minds|s∈VG,ϑ=Maxds|s∈VG; mijG is the edge z1z2∈EG;s≤t.Milan Randic in 1975 established the concept of Randic index [18–20], which is represented as R−1/2G:(2)R−1/2G=∑z1z2∈EG1dz1dz2.The generalized Randic index is defined as [21–28](3)RαG=∑z1z2∈EG1dz1dz2α.Two indicesM1G,M2G were established by Gutman and Trinajstic defined as follows:(4)M1G=∑z1z2∈EGdz1+dz2,M2G=∑z1z2∈EGdz1dz2.Another form of index is which is known as second Zagreb is define as follows [11, 29–32]:(5)Mm2G=∑z1z2∈G1dz1dz2.The symmetric division index is defined as(6)SDDG=∑z1z2∈EGmindz1,dz2maxdz1,dz2+maxdz1,dz2mindz1,dz2.The harmonic index is defined as(7)HG=∑z1z2∈EG2dz1+dz2.“The inverse sum index is defined as(8)IG=∑z1z2∈EGdz1dz2dz1+dz2.The augmented Zagreb index is defined as(9)AG=∑z1z2∈EGdz1dz2dz1+dz2−23.These indices and deliberations which various researchers laboriously worked on can be seen in [33–37] as authentic references.(10)Dγ=γ∂fγ,ε∂γ,Dε=ε∂fγ,ε∂ε,Sγ=∫fγ,εγdγ,Sε=∫fγ,εεdε,Jhγ,ε=fγ,γ,Qαhγ,ε=γαhγ,ε.


## 2. Computational Results on Topological Indices for Indu-Bala Product of Two PathsPk and Pj When k>j

Our fundamental objective of studying M-polynomial and all its related components is to establish a relationship between various affects of M-polynomials and its related things on the Indu-Bala graph, see Figure1.Figure 1
Indu-Bala graphP9▼P7.


### 2.1. Results

We split vertices and edges degree of the Indu-Bala graph in Table1. Similarly, we split the edge palpitations of points on the Indu-Bala graph in Table 2.Table 1
Number of vertices of different degrees of Indu-Bala graph.
Vertices and edgesTotal no. ofVG and EGVG2j+kEG2jk+2j+k−2Table 2
Points of edge partitioning ofPk▼Pj.
EdgesDeg. of end nodesTotal no. of edgesE1dh=j+1,di=j+24E2dh=j+1,di=k+24jE3dh=j+2,di=j+22k−3E4dh=j+2,di=k+22jk−2E5dh=k+2,di=k+22jTheorem 1.
LetG be a Indu-Bala graph Pk▼Pj, where j≥2,k≥j+2. We have(11)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+1bk+2+2jk−2aj+2bj+2+2jak+2bk+2.Proof.
As in Figure1, now we will compute M-polynomial using the values of Tables 1 and 2:(12)MG;a,b=∑i≤jmijG;aibj,=∑j+1≤j+2mj+1j+2Gaj+1bj+2+∑j+1≤k+2mj+1k+2Gaj+1bk+2+∑j+2≤j+2mj+2j+2Gaj+2bj+2+∑j+2≤k+2mj+2k+2Gaj+2bk+2+∑k+2≤k+2mk+2k+2Gak+2bk+2,=∑uv∈Ej+1,j+2mj+1j+2Gaj+1bj+2+∑uv∈Ej+1,k+2mj+1k+2Gaj+1bk+2+∑uv∈Ej+2,j+2mj+2j+2Gaj+2bj+2+∑uv∈Ej+2,k+2mj+2k+2Gaj+2bk+2+∑uv∈Ek+2,k+2mk+2k+2Gak+2bk+2,=Ej+1,j+2aj+1bj+2+Ej+1,k+2aj+1bk+2+Ej+2,j+2aj+2bj+2+Ej+2,k+2aj+2bk+2+Ek+2,k+2ak+2bk+2,=4aj+1bj+2+4jaj+1bk+2+2k−3aj+1bj+2+2jk−2aj+2bk+2+2jak+2bk+2.Theorem 2.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2. Then,(13)M1G=2jkj+k+8+6k−1.Proof.
(14)MG,a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,G,a,b=fa,b;Dafa,b=a∂f∂a,∂f∂a=∂∂a4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,a∂f∂a=4j+1aj+1bj+2+4jj+1aj+1bk+2+2j+2k−3aj+2bj+2+2jj+2k−2aj+2bk+2+2jk+2ak+2bk+2;Dbfa,b=b∂f∂b,∂f∂b=∂∂b4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,b∂f∂b=4j+2aj+1bj+2+4jk+2aj+1bk+2+2j+2k−3aj+2bj+2+2jk+2k−2aj+2bk+2+2jk+2ak+2bk+2,M1G=Da+Dbfa,b|a=b=1=2jkj+k+8+6k−1.Theorem 3.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(15)M2G=8k−j+j22k2+6k−6+2jk10+3k−16.Proof.
Suppose(16)MG,a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,Dafa,b=4j+1aj+1bj+2+4jj+1aj+1bk+2+2j+2k−3aj+2bj+2+2jj+2k−2aj+2bk+2+2jk+2ak+2bk+2,DbDafa,b=4j+1j+2aj+1bj+2+4jj+1k+2aj+1bk+2+2j+22k−3aj+2bj+2+2jj+2k+2k−2aj+2bk+2+2jk+2k+2ak+2bk+2,M2G=DbDafa,b|a=b=1=4j+1j+2+4jj+1k+2+2j+22k−3+2jj+2k2−4+2jk+22,M2G=8k−j+j22k2+6k−6+2jk10+3k−16.Theorem 4.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then (Figures 2 and 3),(17)M2mG=2j4+j32k2−2k+4+j22k2−2k+8+j2k3+8k2−8k+6+4k+12jj+12k+12.Figure 2
(a) First Zagreb index forj≤k. (b) Second Zagreb index for j≤k.
(a)(b)Figure 3
(a) Randic index forj≤k and α=1. (b) Modified second Zagreb index for j≤k and α=1.
(a)(b)Proof.
Suppose(18)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,M2mG=SaSbfa,b|a=b=1,Sa=∫0afu,budu,Sb=∫0bfa,uudu,fu,bu=4ujbj+2+4uujbk+2+2k−3uj+1bj+2+2uk−2uj+1bk+2+2uuk+1bk+2,∗,Sa=4jaj+1bj+2+4aj+1bk+2+2k−3j+1aj+2bj+2+2jk−2j+1aj+2bk+2+2jk+1ak+2bk+2,∫0afa,uudt=4jaj+1uj+1+4aj+1uk+1+2k−3j+1aj+2uj+1+2jk−2j+1aj+2uk+1+2jk+1ak+2uk+1,SbSafa,b=∫0bSafa,uudu,SbSafa,b=4jj+1aj+1bj+2+4k+1aj+1bk+2+2k−3j+12aj+2bj+2+2jk−2j+1k+1aj+2bk+2+2jk+12ak+2bk+2,M2mG=SbSafa,b|a=b=1=4jj+1+4k+1+2k−3j+12+2jk−2j+1k+1+2jk+12,=2j4+j32k2−2k+4+j22k2−2k+8+j2k3+8k2−8k+6+4k+12jj+12k+12.Theorem 5.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(19)RαG=4j+1αj+2α+jk+2α+2j+2αk−3j+2α+2jk−2k+2α+2jk+22α.Proof.
Suppose(20)MG;a;b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,RαG=DaαDbαfa,b|b=a=1,Dbαfa,b=42+jαaj+1b2+j+4j2+kαaj+1b2+k+2−3+kj+2αaj+2bj+2+2j−2+k2+kαaj+2b2+k+2j2+kαak+2b2+k,DaαDbαfa,b=4j+1αj+2αaj+1bj+2+4jj+1αk+2αaj+1bk+2+2k−3j+22αaj+2bj+2+2j−2+k2+jα2+kαaj+2bk+2+2jk+22αak+2bk+2,RαG=DaαDbαfa,b|a=b=1=4j+1αj+2α+4jj+1αk+2α+2−3+k2+j2α+2j−2+k2+jα2+kα+2j2+k2α,RαG=41+jα2+jα+j2+kα+22+jα−3+k2+jα+2j−2+kk+2α+2j2+k2α.Theorem 6.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2. Then,(21)RRαG=41+jα1jα+k−32j+1α+j−2+k21+kα+41+kαj2k+1α+1.Proof.
Suppose(22)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,RRαG=SaαSbαfa,b|a=b=1,Sb=4j+1aj+1bj+2+4jk+1aj+1bk+2+2k−3j+1aj+2bj+2+2jk−2k+1aj+2bk+2+2jk+1ak+2bk+2,Sbα=4j+1αaj+1bj+2+4jk+1αaj+1bk+2+2k−3j+1αaj+2bj+22jk−2k+1αaj+2bk+2+2jk+1αak+2bk+2,SaαSbαfa,b=4jj+1αaj+1bj+2+4k+1αaj+1bk+2+2k−3j+12αaj+2bj+2+2jk−2j+1k+1αaj+2bk+2+2jk+12αak+2bk+2,SaαSbαfa,b|a=b=1=4jj+1α+41+kα+2k−31+j2α+2jk−21+j1+kα+2j1+k2α,RRαG=41+jα1jα+−3+k2j+1α+j−2+k21+kα+41+kαj21+kα+1.Theorem 7.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(23)SSDG=j4k+2+j34k2+3k+2+j26k2+3k−2+4k+13jk+2jj+1k+1.Proof.
Suppose(24)MG;a;b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,SSDG=SbDa+SaDbfa,b|a=b=1,SbDafa,b=4aj+1bj+2+4jj+1k+1aj+1bk+2+2j+2k−3j+1aj+2bj+2+2jj+2k−2k+1aj+2bk+2+2jk+2k+1ak+2bk+2,SaDbfa,b=4j+2jaj+1bj+2+4k+2aj+1bk+2+2j+2k−3j+1aj+2bj+2+2jk+2k−2j+1aj+2bk+2+2jk+2k+1ak+2bk+2;SbDa+SaDbfa,b,=4+4j+2jaj+1bj+2+4jj+1k+1+4k+2aj+1bk+2+2j+2k−3j+1+2j+2k−3j+1aj+2bj+2+2jj+2k−2k+1+2jk+2k−2j+1aj+2bk+2+2jk+2k+1+2jk+2k+1ak+2bk+2,SSDG=SbDa+SaDbfa,b|a=b=1=8j+1j+4k+1j2+k2+j+3k+2+4j+2k−3j+1+jk−2j+1k+1j2+k2+3j+k+4+4jk+2k+1,SSDG=4j2k−4j2+8jk−8j+8jj+1+4j+1j2+k2+j+3k+2+jk−2j2+k2+3j+3k+4+jk+21j+1k+1,=4j2k−4j2+8jk−8j+8jj+1+j3k+2+j24k2+3k+2+j2k2+3k+2+4kk+3+8j+1k+1,SSDG=j4k+2+j34k2+3k+2+j26k2+3k−2+4k+13jk+2jj+1k+1.Theorem 8.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(25)HG=2j3+4j2k2+k+j4k2+10k+12+8k+1jj+1k+1.Proof.
Suppose(26)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,HG=2Sajfa,b|a=1,Safa,b=4jaj+1bj+2+4aj+1bk+2+2k−3j+1aj+2bj+2+2jk−2j+1aj+2bk+2+2jk+1ak+2bk+2,2Sajfa,b=8ja2j+3+8aj+k+3+4k−3j+1a2j+4+4jk−2j+1aj+k+4+4jk+1a2k+4,2Sajfa,b|a=1=8j+8+4k−3j+1+4jk−2j+1+4jk+1,HG=2j3+4j2k2+k+j4k2+10k+12+8k+1jj+1k+1.Theorem 9.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(27)IG=4j5+20j4+j38k2+40+j24k3+14k2−8k+36+j16k2−20k−20+8k+1jj+1k+1.Proof.
Suppose(28)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,IG=SajDaDbfa,b|a=1;DaDbfa,b,=4j+1j+2aj+1bj+2+4jj+1k+2aj+1bk+2+2j+22k−3aj+2bj+2+2jj+2k+2k−2aj+2bk+2+2jk+2k+2ak+2bk+2;SajDaDbfa,b,=4j+1j+2ja2j+3+4j+1k+2aj+k+3+2j+22k−3j+1a2j+4+2jj+2k+2k−2j+1aj+k+4+2jk+2k+2k+1a2k+4;SajDaDbfa,b|a=1,=4j+1j+2j+4j+1k+2+2j+22k−3j+1+2jj+2k2−4j+1+2jk+2k+2k+1,IG=4j5+20j4+j38k2+40+j24k3+14k2−8k+36+j16k2−20k−20+8k+1jj+1k+1.Theorem 10.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(29)IG=2j+232j+132j+13+k−3j+232j+23+2jk+23j+k+132j+13+k−2j+23.Proof.
Suppose(30)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,AG=Sa3Q−2jDa3Db3fa,b|a=1,DaαDbαfa,b=4j+1αj+2αaj+1bj+2+4jj+1αk+2αaj+1bk+2+2k−3j+22αaj+2bj+2+2jk−2j+2αk+2αaj+2bk+2,Da3Db3fa,b=4j+13j+23aj+1bj+2+4jj+13k+23aj+1bk+2+2k−3j+26aj+2bj+2+2jk−2j+23k+23aj+2bk+2,JDa3Db3fa,b=4j+13j+23a2j+3+4jj+13k+23aj+k+3+2k−3j+26a2j+4+2jk−2j+23k+23aj+k+4;Q−2jDa3Db3fa,b,=4j+13j+23a2j+1+4jj+13k+23aj+k+1+2k−3j+26a2j+2+2jk−2j+23k+23aj+k+2;Sa3Q−2jDa3Db3fa,b,=4j+13j+232j+13a2j+1+4jj+13k+23j+k+13aj+k+1+2k−3j+262j+23a2j+2+2jk−2j+23k+23j+k+13aj+k+2;Sa3Q−2jDa3Db3fa,b|a=1=4j+13j+232j+13+4jj+13k+23j+k+13+2k−3j+262j+23+2jk−2j+23k+23j+k+13,AG=2j+232j+132j+13+k−3j+232j+23+2jk+23j+k+132j+13+k−2j+23.


## 2.1. Results

We split vertices and edges degree of the Indu-Bala graph in Table1. Similarly, we split the edge palpitations of points on the Indu-Bala graph in Table 2.Table 1
Number of vertices of different degrees of Indu-Bala graph.
Vertices and edgesTotal no. ofVG and EGVG2j+kEG2jk+2j+k−2Table 2
Points of edge partitioning ofPk▼Pj.
EdgesDeg. of end nodesTotal no. of edgesE1dh=j+1,di=j+24E2dh=j+1,di=k+24jE3dh=j+2,di=j+22k−3E4dh=j+2,di=k+22jk−2E5dh=k+2,di=k+22jTheorem 1.
LetG be a Indu-Bala graph Pk▼Pj, where j≥2,k≥j+2. We have(11)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+1bk+2+2jk−2aj+2bj+2+2jak+2bk+2.Proof.
As in Figure1, now we will compute M-polynomial using the values of Tables 1 and 2:(12)MG;a,b=∑i≤jmijG;aibj,=∑j+1≤j+2mj+1j+2Gaj+1bj+2+∑j+1≤k+2mj+1k+2Gaj+1bk+2+∑j+2≤j+2mj+2j+2Gaj+2bj+2+∑j+2≤k+2mj+2k+2Gaj+2bk+2+∑k+2≤k+2mk+2k+2Gak+2bk+2,=∑uv∈Ej+1,j+2mj+1j+2Gaj+1bj+2+∑uv∈Ej+1,k+2mj+1k+2Gaj+1bk+2+∑uv∈Ej+2,j+2mj+2j+2Gaj+2bj+2+∑uv∈Ej+2,k+2mj+2k+2Gaj+2bk+2+∑uv∈Ek+2,k+2mk+2k+2Gak+2bk+2,=Ej+1,j+2aj+1bj+2+Ej+1,k+2aj+1bk+2+Ej+2,j+2aj+2bj+2+Ej+2,k+2aj+2bk+2+Ek+2,k+2ak+2bk+2,=4aj+1bj+2+4jaj+1bk+2+2k−3aj+1bj+2+2jk−2aj+2bk+2+2jak+2bk+2.Theorem 2.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2. Then,(13)M1G=2jkj+k+8+6k−1.Proof.
(14)MG,a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,G,a,b=fa,b;Dafa,b=a∂f∂a,∂f∂a=∂∂a4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,a∂f∂a=4j+1aj+1bj+2+4jj+1aj+1bk+2+2j+2k−3aj+2bj+2+2jj+2k−2aj+2bk+2+2jk+2ak+2bk+2;Dbfa,b=b∂f∂b,∂f∂b=∂∂b4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,b∂f∂b=4j+2aj+1bj+2+4jk+2aj+1bk+2+2j+2k−3aj+2bj+2+2jk+2k−2aj+2bk+2+2jk+2ak+2bk+2,M1G=Da+Dbfa,b|a=b=1=2jkj+k+8+6k−1.Theorem 3.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(15)M2G=8k−j+j22k2+6k−6+2jk10+3k−16.Proof.
Suppose(16)MG,a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,Dafa,b=4j+1aj+1bj+2+4jj+1aj+1bk+2+2j+2k−3aj+2bj+2+2jj+2k−2aj+2bk+2+2jk+2ak+2bk+2,DbDafa,b=4j+1j+2aj+1bj+2+4jj+1k+2aj+1bk+2+2j+22k−3aj+2bj+2+2jj+2k+2k−2aj+2bk+2+2jk+2k+2ak+2bk+2,M2G=DbDafa,b|a=b=1=4j+1j+2+4jj+1k+2+2j+22k−3+2jj+2k2−4+2jk+22,M2G=8k−j+j22k2+6k−6+2jk10+3k−16.Theorem 4.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then (Figures 2 and 3),(17)M2mG=2j4+j32k2−2k+4+j22k2−2k+8+j2k3+8k2−8k+6+4k+12jj+12k+12.Figure 2
(a) First Zagreb index forj≤k. (b) Second Zagreb index for j≤k.
(a)(b)Figure 3
(a) Randic index forj≤k and α=1. (b) Modified second Zagreb index for j≤k and α=1.
(a)(b)Proof.
Suppose(18)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,M2mG=SaSbfa,b|a=b=1,Sa=∫0afu,budu,Sb=∫0bfa,uudu,fu,bu=4ujbj+2+4uujbk+2+2k−3uj+1bj+2+2uk−2uj+1bk+2+2uuk+1bk+2,∗,Sa=4jaj+1bj+2+4aj+1bk+2+2k−3j+1aj+2bj+2+2jk−2j+1aj+2bk+2+2jk+1ak+2bk+2,∫0afa,uudt=4jaj+1uj+1+4aj+1uk+1+2k−3j+1aj+2uj+1+2jk−2j+1aj+2uk+1+2jk+1ak+2uk+1,SbSafa,b=∫0bSafa,uudu,SbSafa,b=4jj+1aj+1bj+2+4k+1aj+1bk+2+2k−3j+12aj+2bj+2+2jk−2j+1k+1aj+2bk+2+2jk+12ak+2bk+2,M2mG=SbSafa,b|a=b=1=4jj+1+4k+1+2k−3j+12+2jk−2j+1k+1+2jk+12,=2j4+j32k2−2k+4+j22k2−2k+8+j2k3+8k2−8k+6+4k+12jj+12k+12.Theorem 5.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(19)RαG=4j+1αj+2α+jk+2α+2j+2αk−3j+2α+2jk−2k+2α+2jk+22α.Proof.
Suppose(20)MG;a;b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,RαG=DaαDbαfa,b|b=a=1,Dbαfa,b=42+jαaj+1b2+j+4j2+kαaj+1b2+k+2−3+kj+2αaj+2bj+2+2j−2+k2+kαaj+2b2+k+2j2+kαak+2b2+k,DaαDbαfa,b=4j+1αj+2αaj+1bj+2+4jj+1αk+2αaj+1bk+2+2k−3j+22αaj+2bj+2+2j−2+k2+jα2+kαaj+2bk+2+2jk+22αak+2bk+2,RαG=DaαDbαfa,b|a=b=1=4j+1αj+2α+4jj+1αk+2α+2−3+k2+j2α+2j−2+k2+jα2+kα+2j2+k2α,RαG=41+jα2+jα+j2+kα+22+jα−3+k2+jα+2j−2+kk+2α+2j2+k2α.Theorem 6.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2. Then,(21)RRαG=41+jα1jα+k−32j+1α+j−2+k21+kα+41+kαj2k+1α+1.Proof.
Suppose(22)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,RRαG=SaαSbαfa,b|a=b=1,Sb=4j+1aj+1bj+2+4jk+1aj+1bk+2+2k−3j+1aj+2bj+2+2jk−2k+1aj+2bk+2+2jk+1ak+2bk+2,Sbα=4j+1αaj+1bj+2+4jk+1αaj+1bk+2+2k−3j+1αaj+2bj+22jk−2k+1αaj+2bk+2+2jk+1αak+2bk+2,SaαSbαfa,b=4jj+1αaj+1bj+2+4k+1αaj+1bk+2+2k−3j+12αaj+2bj+2+2jk−2j+1k+1αaj+2bk+2+2jk+12αak+2bk+2,SaαSbαfa,b|a=b=1=4jj+1α+41+kα+2k−31+j2α+2jk−21+j1+kα+2j1+k2α,RRαG=41+jα1jα+−3+k2j+1α+j−2+k21+kα+41+kαj21+kα+1.Theorem 7.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(23)SSDG=j4k+2+j34k2+3k+2+j26k2+3k−2+4k+13jk+2jj+1k+1.Proof.
Suppose(24)MG;a;b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,SSDG=SbDa+SaDbfa,b|a=b=1,SbDafa,b=4aj+1bj+2+4jj+1k+1aj+1bk+2+2j+2k−3j+1aj+2bj+2+2jj+2k−2k+1aj+2bk+2+2jk+2k+1ak+2bk+2,SaDbfa,b=4j+2jaj+1bj+2+4k+2aj+1bk+2+2j+2k−3j+1aj+2bj+2+2jk+2k−2j+1aj+2bk+2+2jk+2k+1ak+2bk+2;SbDa+SaDbfa,b,=4+4j+2jaj+1bj+2+4jj+1k+1+4k+2aj+1bk+2+2j+2k−3j+1+2j+2k−3j+1aj+2bj+2+2jj+2k−2k+1+2jk+2k−2j+1aj+2bk+2+2jk+2k+1+2jk+2k+1ak+2bk+2,SSDG=SbDa+SaDbfa,b|a=b=1=8j+1j+4k+1j2+k2+j+3k+2+4j+2k−3j+1+jk−2j+1k+1j2+k2+3j+k+4+4jk+2k+1,SSDG=4j2k−4j2+8jk−8j+8jj+1+4j+1j2+k2+j+3k+2+jk−2j2+k2+3j+3k+4+jk+21j+1k+1,=4j2k−4j2+8jk−8j+8jj+1+j3k+2+j24k2+3k+2+j2k2+3k+2+4kk+3+8j+1k+1,SSDG=j4k+2+j34k2+3k+2+j26k2+3k−2+4k+13jk+2jj+1k+1.Theorem 8.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(25)HG=2j3+4j2k2+k+j4k2+10k+12+8k+1jj+1k+1.Proof.
Suppose(26)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,HG=2Sajfa,b|a=1,Safa,b=4jaj+1bj+2+4aj+1bk+2+2k−3j+1aj+2bj+2+2jk−2j+1aj+2bk+2+2jk+1ak+2bk+2,2Sajfa,b=8ja2j+3+8aj+k+3+4k−3j+1a2j+4+4jk−2j+1aj+k+4+4jk+1a2k+4,2Sajfa,b|a=1=8j+8+4k−3j+1+4jk−2j+1+4jk+1,HG=2j3+4j2k2+k+j4k2+10k+12+8k+1jj+1k+1.Theorem 9.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(27)IG=4j5+20j4+j38k2+40+j24k3+14k2−8k+36+j16k2−20k−20+8k+1jj+1k+1.Proof.
Suppose(28)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,IG=SajDaDbfa,b|a=1;DaDbfa,b,=4j+1j+2aj+1bj+2+4jj+1k+2aj+1bk+2+2j+22k−3aj+2bj+2+2jj+2k+2k−2aj+2bk+2+2jk+2k+2ak+2bk+2;SajDaDbfa,b,=4j+1j+2ja2j+3+4j+1k+2aj+k+3+2j+22k−3j+1a2j+4+2jj+2k+2k−2j+1aj+k+4+2jk+2k+2k+1a2k+4;SajDaDbfa,b|a=1,=4j+1j+2j+4j+1k+2+2j+22k−3j+1+2jj+2k2−4j+1+2jk+2k+2k+1,IG=4j5+20j4+j38k2+40+j24k3+14k2−8k+36+j16k2−20k−20+8k+1jj+1k+1.Theorem 10.
In Indu-Bala graphPk▼Pj, j≥2,k≥j+2; then,(29)IG=2j+232j+132j+13+k−3j+232j+23+2jk+23j+k+132j+13+k−2j+23.Proof.
Suppose(30)MG;a,b=4aj+1bj+2+4jaj+1bk+2+2k−3aj+2bj+2+2jk−2aj+2bk+2+2jak+2bk+2,AG=Sa3Q−2jDa3Db3fa,b|a=1,DaαDbαfa,b=4j+1αj+2αaj+1bj+2+4jj+1αk+2αaj+1bk+2+2k−3j+22αaj+2bj+2+2jk−2j+2αk+2αaj+2bk+2,Da3Db3fa,b=4j+13j+23aj+1bj+2+4jj+13k+23aj+1bk+2+2k−3j+26aj+2bj+2+2jk−2j+23k+23aj+2bk+2,JDa3Db3fa,b=4j+13j+23a2j+3+4jj+13k+23aj+k+3+2k−3j+26a2j+4+2jk−2j+23k+23aj+k+4;Q−2jDa3Db3fa,b,=4j+13j+23a2j+1+4jj+13k+23aj+k+1+2k−3j+26a2j+2+2jk−2j+23k+23aj+k+2;Sa3Q−2jDa3Db3fa,b,=4j+13j+232j+13a2j+1+4jj+13k+23j+k+13aj+k+1+2k−3j+262j+23a2j+2+2jk−2j+23k+23j+k+13aj+k+2;Sa3Q−2jDa3Db3fa,b|a=1=4j+13j+232j+13+4jj+13k+23j+k+13+2k−3j+262j+23+2jk−2j+23k+23j+k+13,AG=2j+232j+132j+13+k−3j+232j+23+2jk+23j+k+132j+13+k−2j+23.


## 3. Computational Results on Topological Indices for Indu-Bala Product of Two PathsPj and Pk When j>k

Our fundamental objective of studying M-polynomial and its related all components is to establish a relationship between various affects of M-polynomials and its related things on the Indu-Bala graph, see Figure4.Figure 4
Indu-Bala graphP7▼P9.


### 3.1. Results

We split vertices and edge degree of the Indu-Bala graph in Table3. Similarly, we split the edge palpitations of points on the Indu-Bala graph in Table 4.Table 3
Number of vertices of different degrees of the Indu-Bala graph.
VerticesTotal no. ofVG and EGVG2j+kEG2jk+2j+k−2Table 4
Points of edge partitioning ofPj▼Pk.
EdgesDeg. of end nodesTotal no. of edgesE1dh=j+2,di=j+22kE2dh=j+2,di=k+14kE3dh=j+2,di=k+2k2j−4E4dh=k+1,di=k+24E5dh=k+2,di=k+22j−6Theorem 11.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(31)MG;a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2.Proof.
As in Figure4, now we will compute M-polynomial using the values of Tables 3 and 4:(32)MG;a,b=∑i≤jmijG;aibj,=∑j+2≤j+2jj+2j+2Gaj+2bj+2+∑j+2≤k+1jj+2k+1Gaj+2bk+1+∑j+2≤k+2jj+2k+2Gaj+2bk+2+∑k+1≤k+2jk+1k+2Gak+1bk+2+∑k+2≤k+2jk+2k+2Gak+2bk+2,=∑uv∈Etj+2,j+2jj+2j+2Gaj+2bj+2+∑uv∈Ej+2,k+1jj+2k+1Gaj+2bk+1+∑uv∈Ej+2,k+2jj+2k+2Gaj+2bk+2+∑uv∈Ek+1,k+2jk+1k+2Gak+1bk+2+∑uv∈Ek+2,k+2jk+2k+2Gak+2bk+2,=Ej+2,j+2aj+2bj+2+Ej+2,k+1aj+2bk+1+Ej+2,k+2aj+2bk+2+Ek+1,k+2ak+1bk+2+Ek+2,k+2ak+2bk+2,=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2.Theorem 12.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(33)M1G=j2jk+2k2+16k+8−12.Proof.
(34)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,G,a,b=fa,b;Dafa,b=a∂f∂a,∂f∂a=∂∂a2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,a∂f∂a=2kj+2aj+2bj+2+4kj+2aj+2bk+1+kj+22j−4aj+2bk+2+4k+1ak+1bk+2+k+22j−6ak+2bk+2;Dbfa,b=b∂f∂b,∂f∂b=∂∂b2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,b∂f∂b=2kj+2aj+2bj+2+4kk+1aj+2bk+1+kk+22j−4aj+2bk+2+4k+2ak+1bk+2+k+22j−6ak+2bk+2;M1G=Da+Dbfa,b|a=b=1,=4kj+2+4kj+k+3+k2j−4j+k+4+42k+3+2k+22j−6,M1G=j2kj+2k2+16k+8−12.Theorem 13.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(35)M2G=j22k2+6k+j6k2+20k+8−2k2−12k−6.Proof.
(36)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,Dafa,b=2kj+2aj+2bj+2+4kj+2aj+2bk+1+kj+22j−4aj+2bk+2+4k+1ak+1bk+2+k+22j−6ak+2bk+2,DbDafa,b=2kj+22aj+2bj+2+4kj+2k+1aj+2bk+1+kj+2k+22j−4aj+2bk+2+4k+1k+2ak+1bk+2+k+222j−6ak+2bk+2,M2G=DbDafa,b|a=b=1=2kj+22+4kj+2k+1+kj+2k+22j−44k+1k+2+k+222j−6,M2G=j22k2+6k+j6k2+20k+8−2k2−12k−6.Theorem 14.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(37)Mm2G=2k4+k32j2−6j+k26j2−2j+k2j3+2j2−10j−6+4j2+8j+4kj+12k+12.Proof.
(38)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,Mm2G=SaSbfa,b|a=b=1,Sa=∫0afx,bxdx;Sb=∫0bfa,xxdx,fx,bx=2kxj+1bj+2+4kxj+1bk+1+k2j−4xj+1bk+2+4xkbk+2+2j−6xk+1bk+2,Sa=2kj+1aj+2bj+2+4kj+1aj+2bk+1+k2j−4j+1aj+2bk+2+4kak+1bk+2+2j−6k+1ak+2bk+2,Saa,xx=2kj+1aj+2xj+1+4kj+1aj+2xk+1+k2j−4j+1aj+2xk+1+4kak+1xk+1+2j−6k+1ak+2xk+1,SbSafa,b=∫0bSafa,xxdx,SbSafa,b=2kj+12aj+2bj+2+4kj+1k+1aj+2bk+2+k2j−4j+1k+1aj+2bk+2+4kk+1ak+1bk+2+2j−6k+12ak+2bk+2,M2mG=SbSafa,b|a=b=1=2kj+12+4kj+1k+1+k2j−4j+1k+1+4kk+1+2j−6k+12,=2k4+k32j2−6j+k26j2−2j+k2j3+2j2−10j−6+4j2+8j+4kj+12k+12.Theorem 15.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(39)RαG=2kj+1αj+2α+2k+1α+kj−2k+2α+2k+2α2k+1α+jk+2α.Proof.
(40)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,RαG=DaαDbαfa,b|a=b=1,Dbαfa,b=2kj+2αaj+2bj+2+4kk+1αaj+2bk+1+k2j−4k+2αaj+2bk+2+4k+2αak+1bk+2+2j−6k+2αak+2bk+2,DaαDbαfa,b=2kj+22αaj+2bj+2+4kj+2αk+1αaj+2bk+1+4k+1αk+2αak+1bk+2+k2j−4j+2αk+2αaj+2bk+2+2j−6k+22αak+2bk+2,RαG=DaαDbαfa,b|a=b=1=2kj+22α+4kj+2αk+1α+k2j−4j+2αk+2α+4k+1αk+2α+2j−6k+22α,RαG=2kj+1αj+2α+2k+1α+kj−2k+2α+2k+2α2k+1α+jk+2α.Theorem 16.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(41)RRαG=2kj+1k2+kαkj+kα+2j+2k+1α+j−2k2+kj+1α+2kαk+12α2k+1α+j−3kα.Proof.
(42)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,RRαG=SaαSbαfa,b|a=b=1,Sb=2kj+1aj+2bj+2+4kkaj+2bk+1+k2j−4k+1aj+2bk+2+4k+1ak+1bk+2+2j−6k+1ak+2bk+2,Sbα=2kj+1αaj+2bj+2+4kkαaj+2bk+1+k2j−4k+1αaj+2bk+2+4k+1αak+1bk+2+2j−6k+1αak+2bk+2,SaαSbαfa,b=2kj+12αaj+2bj+2+4kkj+1αaj+2bk+1+k2j−4j+1k+1αaj+2bk+2+4kk+1αak+1bk+2+2j−6k+12αak+2bk+2,SaαSbαfa,b|a=b=1=2kj+12α+4kkj+1α+k2j−4j+1k+1α+4kk+1α+2j−6k+12α,RRαG=2kj+1α1j+1α+2kα+j−2j+1α+2k+1α2kα+j−3k+1α,RRαG=2kj+1k2+kαkj+kα+2j+2k+1α+j−2k2+kj+1α+2kαk+12α2k+1α+j−3kα.Theorem 17.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(43)SSDG=4j+14j2+k2+jk+3j+3k+2+1kj+1k+12k2j3+j26k2+8k+j2k4+6k3−28k2+8k+16−12k3−28k2−10k+16.Proof.
(44)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,SSDG=SbDa+SaDbfa,b|a=b=1,SbDafa,b=2kj+2j+1aj+2bj+2+4kj+2kaj+2bk+1+k2j−4j+2k+1aj+2bk+2+4k+1k+1ak+1bk+2+2j−6k+2k+1ak+2bk+2,SaDbfa,b=2kj+2j+1aj+2bj+2+4kk+1j+1aj+2bk+1+kk+22j−4j+1aj+2bk+2+4k+2kak+1bk+2+k+22j−6k+1ak+2bk+2;SbDa+SaDbfa,b,=4kj+2j+1aj+2bj+2+k2j−4j2+k2+3j+3k+4j+1k+1aj+2bk+2+4j2+k2+3j+k+2j+1aj+2bj+1+8k+1kak+1bk+2+4j−3k+2k+1ak+2bk+2,SSDG=SbDa+SaDbfa,b|a=b=1=4kj+2j+1+k2j−4j2+k2+3j+3k+4j+1k+1+4j2+k2+3j+k+2j+1+8k+1k+4j−3k+2k+1,SSDG=4j+14j2+k2+jk+3j+3k+2+1kj+1k+12k2j3+j26k2+8k+j2k4+6k3−28k2+8k+16−12k3−28k2−10k+16.Theorem 18.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(45)HG=4j+7k3+k2+kj2+2−k+2kj+1k+1.Proof.
(46)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,HG=2Sajfa,b|a=1,Safa,b=2kj+1aj+2bj+2+4kj+1aj+2bk+1+k2j−4j+1aj+2bk+2+4kak+1bk+2+2j−6k+1ak+2bk+2,2Sajfa,b=4kj+1a2j+4+8kj+1aj+k+3+k4j−8j+1aj+k+4+8ka2k+3+4j−12k+1a2k+42Sajfa,b|a=1=4kj+1+8kj+1+k4j−8j+1+8k+4j−12k+1,HG=4j+7k3+k2+kj2+2−k+2kj+1k+1.Theorem 19.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(47)IG=j22k3+10k2+18k+16+j4k3+16k2+12k−2k2−12k−16j+1k+1.Proof.
(48)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,IG=SajDaDbfa,b|a=1;DaDbfa,b,=2kj+22aj+2bj+2+4kj+2k+1aj+2bk+1+kj+2k+22j−4aj+2bk+2+4k+1k+2ak+1bk+2+k+222j−6ak+2bk+2;SajDaDbfa,b,=2kj+22j+1a2j+4+4kj+2k+1j+1aj+k+3+2kk+2j+2j−2j+1aj+k+4+4k+1k+2k+1a2k+3+2j−6k+22k+1a2k+4;SajDaDbfa,b|a=1,=2kj+22j+1+4kj+2k+1j+1+2kk+2j+2j−2j+1+4k+1k+2k+1+2j−6k+22k+1,IG=j22k3+10k2+18k+16+j4k3+16k2+12k−2k2−12k−16j+1k+1.Theorem 20.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(49)AG=2kj+23j+232j+13+2k+13j+k3+j−2k+23j+k+13+2k+232k+132k3+j−3k+232j+13.Proof.
(50)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,AG=Sa3Q−2jDa3Db3fa,b|a=1,DaαDbαfa,b=2kj+22αaj+2bj+2+4kj+2αk+1αaj+2bk+1+4k+1αk+2αak+1bk+2+k2j−4j+2αk+2αaj+2bk+2+2j−6k+22αak+2bk+2,Da3Db3fa,b=2kj+26aj+2bj+2+4kj+23k+13aj+2bk+1+4k+13k+23ak+1bk+2+k2j−4j+23k+23aj+2bk+2+2j−6k+26ak+2bk+2,JDa3Db3fa,b=2kj+26a2j+4+4kj+23k+13aj+k+3+4k+13k+23a2k+3+k2j−4j+23k+23aj+k+4+2j−6k+26a2k+4;Q−2jDa3Db3fa,b,=2kj+26a2j+2+4kj+23k+13aj+k+1+4k+13k+23a2k+1+k2j−4j+23k+23aj+k+2+2j−6k+26a2k+2;Sa3Q−2jDa3Db3fa,b,=2kj+262j+13a2j+2+4kj+23k+13j+k3aj+k+1+4k+13k+232k3a2k+1+k2j−4j+23k+23j+k+13aj+k+2+2j−6k+262k+13a2k+2;Sa3Q−2jDa3Db3fa,b|a=1,=2kj+262j+13+4kj+23k+13j+k3+4k+13k+232k3+2j−6k+262k+13+k2j−4j+23k+23j+k+13,AG=2kj+23j+232j+13+2k+13j+k3+j−2k+23j+k+13+2k+232k+132k3+j−3k+232j+13.


## 3.1. Results

We split vertices and edge degree of the Indu-Bala graph in Table3. Similarly, we split the edge palpitations of points on the Indu-Bala graph in Table 4.Table 3
Number of vertices of different degrees of the Indu-Bala graph.
VerticesTotal no. ofVG and EGVG2j+kEG2jk+2j+k−2Table 4
Points of edge partitioning ofPj▼Pk.
EdgesDeg. of end nodesTotal no. of edgesE1dh=j+2,di=j+22kE2dh=j+2,di=k+14kE3dh=j+2,di=k+2k2j−4E4dh=k+1,di=k+24E5dh=k+2,di=k+22j−6Theorem 11.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(31)MG;a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2.Proof.
As in Figure4, now we will compute M-polynomial using the values of Tables 3 and 4:(32)MG;a,b=∑i≤jmijG;aibj,=∑j+2≤j+2jj+2j+2Gaj+2bj+2+∑j+2≤k+1jj+2k+1Gaj+2bk+1+∑j+2≤k+2jj+2k+2Gaj+2bk+2+∑k+1≤k+2jk+1k+2Gak+1bk+2+∑k+2≤k+2jk+2k+2Gak+2bk+2,=∑uv∈Etj+2,j+2jj+2j+2Gaj+2bj+2+∑uv∈Ej+2,k+1jj+2k+1Gaj+2bk+1+∑uv∈Ej+2,k+2jj+2k+2Gaj+2bk+2+∑uv∈Ek+1,k+2jk+1k+2Gak+1bk+2+∑uv∈Ek+2,k+2jk+2k+2Gak+2bk+2,=Ej+2,j+2aj+2bj+2+Ej+2,k+1aj+2bk+1+Ej+2,k+2aj+2bk+2+Ek+1,k+2ak+1bk+2+Ek+2,k+2ak+2bk+2,=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2.Theorem 12.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(33)M1G=j2jk+2k2+16k+8−12.Proof.
(34)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,G,a,b=fa,b;Dafa,b=a∂f∂a,∂f∂a=∂∂a2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,a∂f∂a=2kj+2aj+2bj+2+4kj+2aj+2bk+1+kj+22j−4aj+2bk+2+4k+1ak+1bk+2+k+22j−6ak+2bk+2;Dbfa,b=b∂f∂b,∂f∂b=∂∂b2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,b∂f∂b=2kj+2aj+2bj+2+4kk+1aj+2bk+1+kk+22j−4aj+2bk+2+4k+2ak+1bk+2+k+22j−6ak+2bk+2;M1G=Da+Dbfa,b|a=b=1,=4kj+2+4kj+k+3+k2j−4j+k+4+42k+3+2k+22j−6,M1G=j2kj+2k2+16k+8−12.Theorem 13.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(35)M2G=j22k2+6k+j6k2+20k+8−2k2−12k−6.Proof.
(36)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,Dafa,b=2kj+2aj+2bj+2+4kj+2aj+2bk+1+kj+22j−4aj+2bk+2+4k+1ak+1bk+2+k+22j−6ak+2bk+2,DbDafa,b=2kj+22aj+2bj+2+4kj+2k+1aj+2bk+1+kj+2k+22j−4aj+2bk+2+4k+1k+2ak+1bk+2+k+222j−6ak+2bk+2,M2G=DbDafa,b|a=b=1=2kj+22+4kj+2k+1+kj+2k+22j−44k+1k+2+k+222j−6,M2G=j22k2+6k+j6k2+20k+8−2k2−12k−6.Theorem 14.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(37)Mm2G=2k4+k32j2−6j+k26j2−2j+k2j3+2j2−10j−6+4j2+8j+4kj+12k+12.Proof.
(38)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,Mm2G=SaSbfa,b|a=b=1,Sa=∫0afx,bxdx;Sb=∫0bfa,xxdx,fx,bx=2kxj+1bj+2+4kxj+1bk+1+k2j−4xj+1bk+2+4xkbk+2+2j−6xk+1bk+2,Sa=2kj+1aj+2bj+2+4kj+1aj+2bk+1+k2j−4j+1aj+2bk+2+4kak+1bk+2+2j−6k+1ak+2bk+2,Saa,xx=2kj+1aj+2xj+1+4kj+1aj+2xk+1+k2j−4j+1aj+2xk+1+4kak+1xk+1+2j−6k+1ak+2xk+1,SbSafa,b=∫0bSafa,xxdx,SbSafa,b=2kj+12aj+2bj+2+4kj+1k+1aj+2bk+2+k2j−4j+1k+1aj+2bk+2+4kk+1ak+1bk+2+2j−6k+12ak+2bk+2,M2mG=SbSafa,b|a=b=1=2kj+12+4kj+1k+1+k2j−4j+1k+1+4kk+1+2j−6k+12,=2k4+k32j2−6j+k26j2−2j+k2j3+2j2−10j−6+4j2+8j+4kj+12k+12.Theorem 15.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(39)RαG=2kj+1αj+2α+2k+1α+kj−2k+2α+2k+2α2k+1α+jk+2α.Proof.
(40)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,RαG=DaαDbαfa,b|a=b=1,Dbαfa,b=2kj+2αaj+2bj+2+4kk+1αaj+2bk+1+k2j−4k+2αaj+2bk+2+4k+2αak+1bk+2+2j−6k+2αak+2bk+2,DaαDbαfa,b=2kj+22αaj+2bj+2+4kj+2αk+1αaj+2bk+1+4k+1αk+2αak+1bk+2+k2j−4j+2αk+2αaj+2bk+2+2j−6k+22αak+2bk+2,RαG=DaαDbαfa,b|a=b=1=2kj+22α+4kj+2αk+1α+k2j−4j+2αk+2α+4k+1αk+2α+2j−6k+22α,RαG=2kj+1αj+2α+2k+1α+kj−2k+2α+2k+2α2k+1α+jk+2α.Theorem 16.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(41)RRαG=2kj+1k2+kαkj+kα+2j+2k+1α+j−2k2+kj+1α+2kαk+12α2k+1α+j−3kα.Proof.
(42)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,RRαG=SaαSbαfa,b|a=b=1,Sb=2kj+1aj+2bj+2+4kkaj+2bk+1+k2j−4k+1aj+2bk+2+4k+1ak+1bk+2+2j−6k+1ak+2bk+2,Sbα=2kj+1αaj+2bj+2+4kkαaj+2bk+1+k2j−4k+1αaj+2bk+2+4k+1αak+1bk+2+2j−6k+1αak+2bk+2,SaαSbαfa,b=2kj+12αaj+2bj+2+4kkj+1αaj+2bk+1+k2j−4j+1k+1αaj+2bk+2+4kk+1αak+1bk+2+2j−6k+12αak+2bk+2,SaαSbαfa,b|a=b=1=2kj+12α+4kkj+1α+k2j−4j+1k+1α+4kk+1α+2j−6k+12α,RRαG=2kj+1α1j+1α+2kα+j−2j+1α+2k+1α2kα+j−3k+1α,RRαG=2kj+1k2+kαkj+kα+2j+2k+1α+j−2k2+kj+1α+2kαk+12α2k+1α+j−3kα.Theorem 17.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(43)SSDG=4j+14j2+k2+jk+3j+3k+2+1kj+1k+12k2j3+j26k2+8k+j2k4+6k3−28k2+8k+16−12k3−28k2−10k+16.Proof.
(44)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,SSDG=SbDa+SaDbfa,b|a=b=1,SbDafa,b=2kj+2j+1aj+2bj+2+4kj+2kaj+2bk+1+k2j−4j+2k+1aj+2bk+2+4k+1k+1ak+1bk+2+2j−6k+2k+1ak+2bk+2,SaDbfa,b=2kj+2j+1aj+2bj+2+4kk+1j+1aj+2bk+1+kk+22j−4j+1aj+2bk+2+4k+2kak+1bk+2+k+22j−6k+1ak+2bk+2;SbDa+SaDbfa,b,=4kj+2j+1aj+2bj+2+k2j−4j2+k2+3j+3k+4j+1k+1aj+2bk+2+4j2+k2+3j+k+2j+1aj+2bj+1+8k+1kak+1bk+2+4j−3k+2k+1ak+2bk+2,SSDG=SbDa+SaDbfa,b|a=b=1=4kj+2j+1+k2j−4j2+k2+3j+3k+4j+1k+1+4j2+k2+3j+k+2j+1+8k+1k+4j−3k+2k+1,SSDG=4j+14j2+k2+jk+3j+3k+2+1kj+1k+12k2j3+j26k2+8k+j2k4+6k3−28k2+8k+16−12k3−28k2−10k+16.Theorem 18.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(45)HG=4j+7k3+k2+kj2+2−k+2kj+1k+1.Proof.
(46)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,HG=2Sajfa,b|a=1,Safa,b=2kj+1aj+2bj+2+4kj+1aj+2bk+1+k2j−4j+1aj+2bk+2+4kak+1bk+2+2j−6k+1ak+2bk+2,2Sajfa,b=4kj+1a2j+4+8kj+1aj+k+3+k4j−8j+1aj+k+4+8ka2k+3+4j−12k+1a2k+42Sajfa,b|a=1=4kj+1+8kj+1+k4j−8j+1+8k+4j−12k+1,HG=4j+7k3+k2+kj2+2−k+2kj+1k+1.Theorem 19.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(47)IG=j22k3+10k2+18k+16+j4k3+16k2+12k−2k2−12k−16j+1k+1.Proof.
(48)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,IG=SajDaDbfa,b|a=1;DaDbfa,b,=2kj+22aj+2bj+2+4kj+2k+1aj+2bk+1+kj+2k+22j−4aj+2bk+2+4k+1k+2ak+1bk+2+k+222j−6ak+2bk+2;SajDaDbfa,b,=2kj+22j+1a2j+4+4kj+2k+1j+1aj+k+3+2kk+2j+2j−2j+1aj+k+4+4k+1k+2k+1a2k+3+2j−6k+22k+1a2k+4;SajDaDbfa,b|a=1,=2kj+22j+1+4kj+2k+1j+1+2kk+2j+2j−2j+1+4k+1k+2k+1+2j−6k+22k+1,IG=j22k3+10k2+18k+16+j4k3+16k2+12k−2k2−12k−16j+1k+1.Theorem 20.
In Indu-Bala graphPj▼Pk, j≥2,k≥j+2; then,(49)AG=2kj+23j+232j+13+2k+13j+k3+j−2k+23j+k+13+2k+232k+132k3+j−3k+232j+13.Proof.
(50)MG,a,b=2kaj+2bj+2+4kaj+2bk+1+k2j−4aj+2bk+2+4ak+1bk+2+2j−6ak+2bk+2,AG=Sa3Q−2jDa3Db3fa,b|a=1,DaαDbαfa,b=2kj+22αaj+2bj+2+4kj+2αk+1αaj+2bk+1+4k+1αk+2αak+1bk+2+k2j−4j+2αk+2αaj+2bk+2+2j−6k+22αak+2bk+2,Da3Db3fa,b=2kj+26aj+2bj+2+4kj+23k+13aj+2bk+1+4k+13k+23ak+1bk+2+k2j−4j+23k+23aj+2bk+2+2j−6k+26ak+2bk+2,JDa3Db3fa,b=2kj+26a2j+4+4kj+23k+13aj+k+3+4k+13k+23a2k+3+k2j−4j+23k+23aj+k+4+2j−6k+26a2k+4;Q−2jDa3Db3fa,b,=2kj+26a2j+2+4kj+23k+13aj+k+1+4k+13k+23a2k+1+k2j−4j+23k+23aj+k+2+2j−6k+26a2k+2;Sa3Q−2jDa3Db3fa,b,=2kj+262j+13a2j+2+4kj+23k+13j+k3aj+k+1+4k+13k+232k3a2k+1+k2j−4j+23k+23j+k+13aj+k+2+2j−6k+262k+13a2k+2;Sa3Q−2jDa3Db3fa,b|a=1,=2kj+262j+13+4kj+23k+13j+k3+4k+13k+232k3+2j−6k+262k+13+k2j−4j+23k+23j+k+13,AG=2kj+23j+232j+13+2k+13j+k3+j−2k+23j+k+13+2k+232k+132k3+j−3k+232j+13.


## 4. Graphical Results and Their Discussion

In this section, we present some graphical results which are related to M-polynomial and their degree-based indices for the Indu-Bala product of two paths when one path is greater than another path and vice versa. We have used different values ofk and j and drawn their respective graphs as shown in Figure 5(a) (inverse Randic index for j≥k and α=1) and Figure 5(b) (symmetric division index for j≥k and α=1). We have observed from Figure 6(a) (harmonic index for j≥k and α=1) and Figure 6(b) (inverse sum index for j≥k and α=1) that the overall structure of indices increase with the increase of the value of j and k.Figure 5
(a) Inverse Randic index forj≥k and α=1. (b) Symmetric division index for j≥k and α=1.
(a)(b)Figure 6
(a) Harmonic index forj≥k and α=1. (b) Inverse sum index for j≥k and α=1.
(a)(b)


## 5. Conclusions

It is important to research the network through charts, and topological indicators are important for understanding its basic topology. This type of research finds global application in computer science, networks, and communication systems, which uses various indexes based on graph invariance to consider some stimulation summary. The Indu-Bala networks we studied in this paper are used to optimize (minimized) the operational cost of the network and find the shortest linkage between the connectors.In this article, we present some product for M-polynomial and nine different degree-based topological descriptors as discussed above for the Indu-Bala product of two paths.



---

*Source: 6281596-2021-11-28.xml*