Polynomials of Degree-Based Indices for Swapped Networks Modeled by Optical Transpose Interconnection System

The Optical Transpose Interconnection System (OTIS) has applications in parallel processing, distributed processing, routing, and networks. It is used for efficient usage of multiple parallel algorithms or parallel systems, with different global interconnections in a network as it is an optoelectronic (combination of light signals and electronics). In chemical graph theory, topological indices are used to study characteristics of the chemical structures or biological activities. Topological indices are sometimes studied with the assistance to their polynomial. In this article, polynomials of degree-based topological indices for OTIS and swapped networks have been studied. Results can be used to compute any degree-based topological polynomials for OTIS swapped network.


I. INTRODUCTION AND PRELIMINARY RESULTS
Graph theory has been proved as a vast field by solving problems in multiple fields, like chemistry, physics, computer science, statistics, robotics, networks and routing. Graph can be represented numerically in different forms of data container like matrices, vectors and polynomials. Different real life problems can be represented with the help of a graph. Graph theory provides further operations to find the solution of a problem. Chemical graph theory has further branches quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR), which are essential part in studying the characteristics or chemical properties of molecules and atoms [1]- [4].
Topological index is a single value used to represent the characteristics of the graph. It is invariant under graph automorphism. Topological indices have played an important part in the study of chemical properties under the branches of graph theory QSPR and QSAR. They are used to correlate biological activity or other properties of molecules with their chemical structure. Weiner Index was the first topological index, introduced by Harry Weiner in 1947. Topological indices can be categorized on the bases of their calcula-The associate editor coordinating the review of this manuscript and approving it for publication was Donghyun Kim . tion mechanism. Degree based topological indices involves degree of vertices of graph in the calculation. Randić index, Zagreb index, harmonic index, atom bond connectivity and geometric-arithmetic index are some known degree-based topological indices and polynomials [5]- [17].
OTIS is an optoelectronic. In a network, it provides competency in both optical and electronic technologies. Multiple groups are connected efficiently, electronic connections are used within the same group while optical links are used to communicate between different groups. Multiple algorithms are used for routing, image processing, parallel processing, matrix multiplication (hybrid network model), sorting, selecting and fourier transform (sound and signals) [18]- [20]. A network can be represented graphically. Servers and processors can be represented by vertices while the connections between them can be represented by edges. The number of links on servers or processors are degree of vertices. The maximum distance between two network heads is grid diameter [21]- [23].

II. DEGREE-BASED INDICES AND THEIR POLYNOMIALS
Let G be a graph with the vertex set V (G) and the edge set E(G). The degree d υ of a vertex υ ∈ V (G) is the number of neighbours of υ. The most general indices based on degrees VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ are the general Randić index of a graph G, the general sum-connectivity index and the generalized Zagreb index Note that the third redefined Zagreb index is defined as the harmonic index is defined as the third Zagreb index the fourth Zagreb index and the fifth Zagreb index Let us introduce a general invariant for polynomials of above mentioned topological indices.
where ϕ(d υ , d ν ) is a function of d υ and d ν such that where α is a positive integer, then P(G, x) is the general sum-connectivity polynomial of G. Furthermore, P(G, x) is the first Zagreb polynomial for α = 1 and the hyper-Zagreb polynomial for α = 2.
where α is a positive integer and β is a non-negative integer, then P(G, x) is the generalized Zagreb polynomial of G. Moreover, P(G, x) is the forgotten polynomial if α = 2 and β = 0.
Note that the harmonic polynomial is defined differently from the other polynomials.
is the fifth Zagreb polynomial of G. So the general Randić polynomial of any graph G is defined as the general sum-connectivity polynomial is the generalized Zagreb polynomial of any graph G, the third redefined Zagreb polynomial is defined as the harmonic polynomial is the third Zagreb polynomial the fourth Zagreb polynomial and the fifth Zagreb polynomial

III. OPTICAL TRANSPOSE INTERCONNECTION SYSTEM SWAPPED NETWORK
Considered a graph G having vertex set V (G) and edge set E(G), OITS swapped network O G can be defined as below: [24]. For the mutual OTIS network O G , the graph G is called the factor of the graph or grid. If there are m primary network G. So, O G consists of a separate subnet from the m node groups are called, and they are similar to G [24]. The node name (x, y) in O G select the y node handle in the group [18]- [20], [25]- [27]. Next some some degree based topological polynomials of swapped networks are calculated. For a given path P m on m vertices and O P m as its OTIS swapped network with basis network P m is shown in Figure 1.

IV. RESULTS FOR OTIS SWAPPED NETWORK O P m
We present results, which can be used to compute any degree-based topological polynomials. Our results generalize known results in the area. We give exact values of the most well-known degree-based polynomials for optical transpose interconnection system O P m . Vetrík [28] introduced a new method to calculate the topological indices and also in [29], we follow the same technique in this paper. Let us give a formula, which can be used to obtain any polynomial of indices based on degrees for optical transpose interconnection system O P m .
This means that the set V j contains the vertices of degree j. The set of vertices with respect to their degrees are as follows: Let us divide the edges of O P m into partition sets according to the degree of its end vertices. Let 3,3 . The number of edges incident to one vertex of degree 1 and another vertex of degree 3 is 2, so | 1,3 | = 2. The number of edges incident to two vertices of degree 2 is 3, so | 2,2 | = 3. The number of edges incident to one vertex of degree 2 and other vertex of degree 3 are 6m − 14, so | 2,3 | = 6m − 14. Now, the remaining number of edges are those edges which are incident to two vertices of degree 3, i.e 3,3 . Now we present polynomials of the best-known degree based polynomials of optical transpose interconnection system in the following theorem.
Theorem 1: For the optical transpose interconnection sys- In the next theorem, we determined general sumconnectivity polynomial, first Zagreb polynomial and hyper-Zagreb polynomial of the optical transpose interconnection system O P m .
Theorem 2: For the optical transpose interconnection system O P m , we have the general sum-connectivity polynomial of O P m , For α = 1, the first Zagreb polynomial is + 9 x 36 . In the following theorem, we determined generalized Zagreb polynomial and forgotten polynomial of the optical transpose interconnection system O P m .
Theorem 3: For the optical transpose interconnection system O P m , we have the generalized Zagreb polynomial of O P m , 18 . In the following theorem, we determined the third redefined Zagreb polynomial and harmonic polynomial of the optical transpose interconnection system O P m .
In the following theorem, we determined the third Zagreb polynomial, fourth Zagreb polynomial and fifth Zagreb polynomial of the optical transpose interconnection system O P m .
Theorem 5: For the optical transpose interconnection system O P m , we have the third Zagreb polynomial of O P m , the fourth Zagreb polynomial of O P m , Thus by Lemma 1,

V. OTIS SWAPPED NETWORK O K m
The complete graph denoted by K m with m vertices and O K m be the OTIS swapped network for O K 4 as example shown in Figure 2.
This means that the set V j contains the vertices of degree j. The set of vertices with respect to their degrees are as follows: Hence, x λ(m,m) . After Simplification, we get }. Theorem 6: For the optical transpose interconnection sys- − m) α and λ(m, m) = (m 2 ) α . Thus by Lemma 2, For α = 1, the second Zagreb polynomial is In the next theorem, we determined general sumconnectivity polynomial, first Zagreb polynomial and hyper-Zagreb polynomial of the optical transpose interconnection system swapped network O K m .
Theorem 7: For the optical transpose interconnection system swapped network O K m , we have the general sum-connectivity polynomial of O K m , }. the hyper-Zagreb polynomial of O K m , Proof 9: For χ α (O K m , x) which is the general sumconnectivity polynomial of O K m , we have λ(d ε , d ν ) = (d ε + d ν ) α , therefore λ(m, m − 1) = (2m − 1) α and λ(m, m) = (m 2 ) α . Thus by Lemma 2, For α = 2, the hyper-Zagreb polynomial is In the following theorem, we determined generalized Zagreb polynomial and forgotten polynomial of the optical transpose interconnection system swapped network O K m .
}. For α = 2, β = 0, the forgotten polynomial is }. In the following theorem, we determined the third redefined Zagreb polynomial and harmonic polynomial of the optical transpose interconnection system swapped network O K m .
Theorem 9: For the optical transpose interconnection system swapped network O K m , we have the third redefined Zagreb polynomial of O K m , −m)) α and λ(m, m) = x (2m 3 ) α . Thus by Lemma 2, In the following theorem, we determined the third Zagreb polynomial, fourth Zagreb polynomial and fifth Zagreb polynomial of the optical transpose interconnection system swapped network O K m .

VI. CONCLUSION
Optical Transpose Interconnection Systems (OTIS) swapped networks are optoelectronic and have been used in efficient parallel processing and services in large global networks. Topological indices are often studied with the assistance of their polynomials. Formulae for degree-based topological polynomials for Optical Transpose Interconnection Systems swapped network have been derived. Results can be used to compute any degree-based topological polynomials for OTIS swapped network. These results will help in the future research of networks, mechanics, computer science, and chemistry.