New Degree-Based Topological Indices of Toroidal Polyhex Graph by Means of M-Polynomial

Graph theory is the principal field of mathematics. In this manuscript, we have discussed the toroidal polyhex graph. Some new indices such as reduced reciprocal randic, arithmetic geometric, SK, SK 1 , SK 2 indices, First Zagrab, the general sum-connectivity, SCI λ , and the forgotten index have been used. We have computed the closed form of topological indices of toroidal polyhex graph via M-Polynomial.


Introduction
Graph theory in mathematics means the study of graphs. Graphs are one of the prime objects of study in discrete mathematics. e graph appears as a set of vertices (nodes or points) connected by edges (arcs or lines). Graphs are mathematical structures of the diagram formed by using model pairwise relation between objects.
ey are found on road maps and constellations when constructing schemes and drawing. Graphs underlie many computer programs that make modern communication and technological processes possible. A chemical graph theory is the mixture of two subjects' chemistry and mathematics. e chemical graph is the topological type of mathematical chemistry [1] which declares in a graph to mathematical modeling of the chemical event. Sometimes mathematical chemistry is also called computer chemistry [2]. Chemical graph is concerned with searching the topological indices associated with the properties of chemical molecules [3].
A graph G(V; E) with vertex set V(G) and edge set E(G) is connected if there exists a connection between any pair of vertices in G. A network directly connected graph having no multiple edges and loops. e degree of a vertex is several vertices that are fastened to the connected vertex by the edges. e rst topological index was used by Wiener [4]. Topological indices work for the success of the quantitative activity and other properties of a molecule that correlate with chemical structure. e connection between atoms shown by various types of topological indices give a good guess of di erent chemical properties of the chemical compound such as boiling point, the heat of formation, evaporation, surface tension, and vapor pressure. e rst topological index was used by Wiener [5]. Topological indices work for the success of the quantitative activity and other properties of a molecule that correlate with chemical structure. e connection between atoms shown by various types of topological indices gives a good guess of di erent chemical properties of the chemical compound such as boiling point, the heat of formation, evaporation, surface tension, and vapor pressure. e topological indices are computed via M-polynomial. Several works are done in this area [6][7][8].

Reduced Reciprocal Randic Index
(1) In 2015 [9], Gutman and Furtula introduced a reduced reciprocal index. e reduced reciprocal randic (RRR) index is a molecular structure descriptor (or more precisely, a topological index), handy for a divine level of enthalpy creation and usual boiling point of isomeric octanes.
Definition 1. e M-polynomial is firstly used in 2015 [8] and is determined as follows: where In last few years, M-polynomial of several graphs is invented [5,[13][14][15][16]. In Table 1, degree-dependent topological indices via M-polynomial are provided where

Toroidal Polyhex Network
Fullerene was published in 1985. New forms of the element carbon (C) were established by Robert C, Richard E. Smalley, and Sir Harold W. K. Fullerene is an allotrope form of carbon whose molecules exist in carbon atoms attached by single and double bonds that can be the form of closed mesh or slightly closed mesh, with a fused ring of five to seven atoms. ese molecules may be hollow spheres, ellipsoid, tube, and many more shapes and sizes. Let en, (1) Reduced reciprocal randic index is as follows: mn,

Topological index Derivation from M(G; x, y)
Reduced reciprocal randic Arithmetic geometric index

Journal of Mathematics
(2) Arithmetic geometric index is as follows: � 3mn.

Conclusion
In this article, we assess the toroidal polyhex graph through the degree-based topological indices. e plot of topological indices of toroidal polyhex is given in Figure 3. e M-polynomial calculated the toroidal polyhex that can help us to understand and recover many degree-based topological indices. ese topological indices play a vital role.

Data Availability
No data were used to support this study.

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