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 graphs and 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 . This study states we have found M-polynomial and degree-based topological indices for Indu-Bala product of two paths and for . We also give some graphical representation of these indices and analyzed them graphically.
1. Introduction
Let be a simple and finite graph of order . We denote the nonempty vertex set by and edge set by . 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 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 of graphs and is obtained from two disjoint copies of the join of and by joining the corresponding vertices in the two copies of .
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.
As in [17], ; is the edge .
Milan Randic in 1975 established the concept of Randic index [18–20], which is represented as :
The generalized Randic index is defined as [21–28]
Two indices were established by Gutman and Trinajstic defined as follows:
Another form of index is which is known as second Zagreb is define as follows [11, 29–32]:
The symmetric division index is defined as
The harmonic index is defined as
“The inverse sum index is defined as
The augmented Zagreb index is defined as
These indices and deliberations which various researchers laboriously worked on can be seen in [33–37] as authentic references.
2. Computational Results on Topological Indices for Indu-Bala Product of Two Paths and When
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 Figure 1.
2.1. Results
We split vertices and edges degree of the Indu-Bala graph in Table 1. Similarly, we split the edge palpitations of points on the Indu-Bala graph in Table 2.
Theorem 1. Let be a Indu-Bala graph , where . We have
Proof. As in Figure 1, now we will compute M-polynomial using the values of Tables 1 and 2:
Theorem 2. In Indu-Bala graph , . Then,
Proof.
Theorem 3. In Indu-Bala graph , ; then,
Proof. Suppose
Theorem 4. In Indu-Bala graph , ; then (Figures 2 and 3),
(a)
(b)
(a)
(b)
Proof. Suppose
Theorem 5. In Indu-Bala graph , ; then,
Proof. Suppose
Theorem 6. In Indu-Bala graph , . Then,
Proof. Suppose
Theorem 7. In Indu-Bala graph , ; then,
Proof. Suppose
Theorem 8. In Indu-Bala graph , ; then,
Proof. Suppose
Theorem 9. In Indu-Bala graph , ; then,
Proof. Suppose
Theorem 10. In Indu-Bala graph , ; then,
Proof. Suppose
3. Computational Results on Topological Indices for Indu-Bala Product of Two Paths and When
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 Figure 4.
3.1. Results
We split vertices and edge degree of the Indu-Bala graph in Table 3. Similarly, we split the edge palpitations of points on the Indu-Bala graph in Table 4.
Theorem 11. In Indu-Bala graph , ; then,
Proof. As in Figure 4, now we will compute M-polynomial using the values of Tables 3 and 4:
Theorem 12. In Indu-Bala graph , ; then,
Proof.
Theorem 13. In Indu-Bala graph , ; then,
Proof.
Theorem 14. In Indu-Bala graph , ; then,
Proof.
Theorem 15. In Indu-Bala graph , ; then,
Proof.
Theorem 16. In Indu-Bala graph , ; then,
Proof.
Theorem 17. In Indu-Bala graph , ; then,
Proof.
Theorem 18. In Indu-Bala graph , ; then,
Proof.
Theorem 19. In Indu-Bala graph , ; then,
Proof.
Theorem 20. In Indu-Bala graph , ; then,
Proof.
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 of and and drawn their respective graphs as shown in Figure 5(a) (inverse Randic index for and ) and Figure 5(b) (symmetric division index for and ). We have observed from Figure 6(a) (harmonic index for and ) and Figure 6(b) (inverse sum index for and ) that the overall structure of indices increase with the increase of the value of and .
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(b)
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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.
Data Availability
There are no data used in this research.
Conflicts of Interest
The authors declare no conflicts of interest.