On Analysis and Computation of Degree-Based Topological Invariants for Cyclic Mesh Network

. Recently, there has been increasing attention on the system network due to its promising applications in parallel hanging architectures such as distributed computing (Day (2004), Day and Al-Ayyoub (2002)). Related networks diﬀer in the circumstances of topology, and the descriptors were freshly examined by Hayat and Imran (2014) and Hayat et al. (2014). Distance-based descriptors, counting-related descriptors, and degree-based descriptors are all examples of topological descriptors. These topological characteristics are linked to chemical features of a substance, such as stability, strain energy, and boiling point. The speciﬁcations for the 1st Zagreb alpha, 1st Zagreb beta, 2nd Zagreb, sum-connectivity, geometric-arithmetic, Randic, harmonic, and atom-bond connectivity indices for mesh networks ( MN m × n ) based on VE and EV degree are discussed in this paper.


Introduction
Cheminformatics is a relatively new field that combines chemistry, mathematics, and information science. Cheminformatics is primarily used to store, index, and retrieve information on chemicals. In index factors, graph theory is very essential. Biological activity is used as an introduction to numerous structural properties of molecules in the study of (QSAR) models. Topological indicators are a fascinating subset of these factors. Topological indices can be calculated using simply nodes (atoms) and edges in a graph representation (chemical bonds) [1,2]. A series of numbers, a polynomial, or a numeric number can all be used to identify a graph. A complete graph is represented by numbers or an array, and for those graphs, these interpretations are supposed to be unique. e topological index is a mathematical term that belongs to a graph and is unaffected by graph automorphism. It identifies the structure of the graph. Degree-based topological indices, counting-related polynomials, and graph indices are some of the most common types of topological indices in [3,4]. A topological index is a function "Top" from Σ to the set of real numbers. Topological indicators in a network are, of course, the number of nodes and links present in the network. Numerous networks having an atomic or molecular structure, such as honeycomb, grid networks, and hexagonal, are comparable. Topological properties of this network are very interesting, which are studied in various aspects in [5,6]. Hexagonal and honeycomb structures have also been identified as important in biological evolution, where intersecting triangles are critical for transmission of aid in societal problems, particularly for the development of collaboration [7,8]. Appropriate research on this topic and further benefit from the new research findings are given in [9][10][11][12]. While working on paraffin breaking, Wiener's approach [13] gave the impression of being a topological description. is identifier is known as the route number. After this, Wiener index is used to remember the route number. In terms of theory and practice, this topological descriptor served as the foundation for the topological index, see [14,15] for details. erefore, the topological lesions in the chemical and quantitative literature are Weiner, Zagreb, and Randic [16,17]. By using the previous degree concepts, all of the above works were completed. In grid history, Chellali et al. [18] developed two additional degree theories, namely, VE degrees and EV degrees, after some time. ese concepts are a twist on the previous degree-based concept.

Preliminaries
In this section, we recall some fundamental definitions concerned to network which is usually represented by N � (P, C), where P is the set of points and C is the set of connections of network. e degree of a point is the number of connections joined to that point. e open neighborhood, indicated as N(v), for a point v, is written as N(v) � u ∈ P|u, v ∈ C in networks. If we add the point v to the set of N(v), we get the same as the closed neighborhood of v, represented by N [v]. e number of connections that are connected to any points from the closed neighbourhood of v is equal to the vertex-edge degree, denoted by ϕ ve (v), specified in [19], of the point v ∈ P. Furthermore, the number of points in the union of the closed neighbourhoods of u and v is equal to the edge-vertex degree of the connection e � uv ∈ C, indicated by ϕ e v(e) specified in [19]. Let N be a simple network and e � uv ∈ C(N). e EV and VE degree topological descriptor-related details can be seen in [18,19].
Consider N to be a basic network in all of the illustrations and v ∈ V(N).
e Zagreb indices for edge-vertex degree are specified as e first Zagreb alpha indices for vertex-edge degree are specified as e first Zagreb beta indices for vertex-edge degree are specified as e second Zagreb indices for vertex-edge degree are specified as e Randic indices for vertex-edge degree are specified as e Randic indices for edge-vertex degree are specified as e atom-bond connectivity indices for vertex-edge degree are specified as e geometric-arithmetic (ve − GA) indices for vertexedge degree are specified as e harmonic (ve − H) indices for vertex-edge degree are specified as e sum-connectivity (ve-χ) indices for vertex-edge degree are specified as

Result for Mesh Network (MN m×n )
In this part, we calculate several mesh network (MN m×n ) topological indices based on EV and VE degree, seen in Figure 1. Let (MN m×n ) be a mesh network. (MN m×n ) has mn points and 2mn − (m + n) connections. Table 1 shows how we partition the set of connections of (MN m×n ) into four components on the behalf of degrees of end points.-Similarly, partition the collection of points into three components based on the degrees of points, as shown in Table 2.
Depending on the EV degree of the links of (MN m×n ) for n ≥ 5, we separate them in Table 3 and partition the points and connections in Table 4 and Table 5 depending on the VE degree of (MN m×n ) for n ≥ 5.

Edge-Vertex Degree-Based
Indices. Now, we will calculate Zagreb and Randic indices for the mesh network (MN m×n ) depending on the EV degree.   Table 3, the Randic index is calculated as follows:

Vertex-Edge Degree-Based
Indices. Now, we will calculate the 1st Zagreb alpha, 1st Zagreb beta, 2nd Zagreb, geometric-arithmetic, sum-connectivity, Randic, harmonic, and atom-bond connectivity indices for the mesh network (MN m×n ) depending on VE. Table 4, the 1st Zagreb alpha index is calculated as follows:  Table 5, the 1st Zagreb beta index is calculated as follows:  Table 5, the 2nd Zagreb index is calculated as follows:  Table 5, the Randic index is calculated as follows: Journal of Chemistry

e Atom-Bond Connectivity Index.
Employing Table 5, we calculate the above said index as follows:

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(m + n) 3.2.6. e Geometric-Arithmetic Index. Employing Table 5, the above said index is calculated as follows:

Conclusion
It's crucial to explore the structure using graphs, and topological indicators are crucial for grasping the network's core topology.
is sort of analysis has a wide range of applications in computer science, where different indexes based on graph invariance are used to evaluate multiple stimulation summaries. Invariants stats are essential factors for analyzing and predicating the features of chemical structures in the quantitative structure-property relationship (QSPR) and quantitative tructure-activity relationship (QSAR) explorations. We offer several finished products for VE degree and EV degree-based indices, such as the indices depending on the vertex-edge degree are the 1st Zagreb alpha, 1st Zagreb beta, 2nd Zagreb, geometric-arithmetic, sum-connectivity, Randic, harmonic, and atom-bond connectivity indices, in this article, for the mesh network (MN m×n ), and EV degree Randic and Zagreb indices. ere will be some who are involved in designing new grids in the future, and we are examining their topological indices in order to grasp their core topology.

Data Availability
e data used to support the findings of this study are cited at relevant places within the text as references.

Conflicts of Interest
e authors declare that they have no conflicts of interest.