Topological Properties of Degree-Based Invariants via M-Polynomial Approach

Chemical graph theory provides a link between molecular properties and a molecular graph.)eM-polynomial is emerging as an efficient tool to recover the degree-based topological indices in chemical graph theory. In this work, we give the closed formulas of redefined first and second Zagreb indices, modified first Zagreb index, nano-Zagreb index, second hyper-Zagreb index, Randić index, reciprocal Randić index, first Gourava index, and product connectivity Gourava index via M-polynomial. We also present the M-polynomial of silicate network and then closed formulas of topological indices are applied on the silicate network.


Introduction
Chemical graph theory (CGT) provides a connection between chemical structure and discrete mathematics. e ordered pair between the vertex set and edge set is considered a graph. In CGT, a graph is the replacement of molecular structure in which vertices are replaced by atoms and edges are considered as bonds. A topological index or invariant is an important tool of CGT that can guess the properties of the chemical species [1]. A mapping I: Υ ⟶ R is called a topological invariant if ∀ G, H ∈ Υ, G and H are isomorphic to each other if and only if I(G) � I(H), where Υ is the collection of molecular structure. Topological indices are a very useful application in structureactivity and structure-property modeling [2]. ese are the best predictor to determine the physical, chemical, and biological behavior of the chemical substances [3].
Without performing any testing, topological indices analyze many physicochemical properties of chemical compounds. Topological indices analyse the physical characteristics such as molar volume, melting point, surface tension, boiling point, molar refraction, and heats of vaporization.
Topological indices also represent the biological behavior of compounds such as pH regulation, stimulation of cell growth, lipophilicity, toxicity, and nutrition. Hence, these topological indices may be helpful to know the chemical and physical characteristics and biological behaviors. e first topological index is the Wiener index which was initiated by Wiener in 1947 [4]. After that, thousands of indices have been designed till now [5]. A degree-dependent topological index for the graph G is defined as (1) where d x , d y � j, k and the total number of edges xy is denoted by m jk . In 2013, Ranjini et al. introduced the new version of Zagreb indices known as redefined second and third Zagreb indices defined as [6] In 2003, the theory of Zagreb indices was reviewed in [7] written by Nikolić et al. and the new index was named as the modified first Zagreb index defined as In 2019, Jahanbani and Shooshtary [8] presented the nano-Zagreb index stated as: Alameri [9] proposed the second hyper Zagreb index in 2021 and defined as A bond-additive topological index was proposed by Randic in 1975 [10]. is index is a frequently examined index among all previously topological descriptors: In 2014, Gutman et al. [11] presented the reciprocal Randić index defined as In 2014, Kulli [12] presented the first Gourava index defined as In 2017, Kulli [13] presented the product connectivity Gourava index and defined as By using different tools, we convert the molecular graph into some algebraic polynomial. Using these polynomials, the structural properties are determined easily [14]. ese topological indices are either calculated directly by their formula or by using the graph polynomials such as M-polynomial. Deutsch and Klawzar defined the M-polynomial in 2015 and gave the nine closed formulas of the topological indices via M-polynomial [15]. After that, a lot of computation on different graphs was done in this area [16,17]. In 2020, Afzal et al. proposed nine more closed formulas of topological indices by using the M-polynomial of the graph [18], and Hussain provided the relationship between the direct form and closed form via M-polynomial in 2021 [19]. In [20] Rajpoot and Selvaganesh presented another five closed forms of topological indices via M-polynomial. Shin et al. presented in 2021 a new set of nine topological indices calculated by the M-polynomial of the graph G [21]. In Table 1, we introduced nine more indices of degree-dependent topological indices via M-polynomial.
M-polynomial for the graph G is defined as Here, ψ � min d x |x ∈ V G and Ψ � max d x |x ∈ V G . Some operators, which are used in Table 1, are defined as

Main Results
In the present section, proofs of closed formulas mentioned in Table 1 are provided.  ). ) represents the M-polynomial for the graph G, then the redefined first Zagreb index is also calculated as Proof. By taking Hence, ) represents the M-polynomial of the graph G, then the redefined second Zagreb index is given by Proof. By taking Journal of Mathematics 3 Hence, ReZG 2 (G) � S u JD u D v M G (u, v)| u�1 .

□ Theorem 3. If M G (u, v) represents the M-polynomial of the graph G, then the modified first Zagreb index is computed by
Proof. By taking

) represents the M-polynomial of the graph G, then the nano-Zagreb index is given by
Proof. By taking

) represents the M-polynomial of the graph G, then the second hyper-Zagreb index is computed as
Proof. By taking Hence, v) represents the M-polynomial of the graph G, then the Randić index is computed as Proof. By taking v) represents the M-polynomial for the graph G, then the reciprocal Randić index is given by Proof. By taking ) represents the M-polynomial of the graph G, then the first Gourava index is computed as Proof. By taking Journal of Mathematics Hence, ) represents the M-polynomial for the graph G, then the product connective Gourava index is computed as PGO( Proof. By taking Hence, PGO(G) � S 1/2 u JS 1/2

Chemical Graph of Silicate Network
e chemical graph of the silicate network (SL n ) is shown in Figure 1, where n is the total number of hexagons present between the center of the network and the boundary of SL n . In Figure 1, blue and red dots represent the vertices having degrees 3 and 6, respectively. e edges having end-degrees (3,3), (3,6), and (6, 6) are represented with yellow, green, and brown lines, respectively. Table 2 shows the vertex partitions, and Table 3 represents the edge partitions of SL n . In this present work, we extract some topological indices via M-polynomial of SL n .
e surface plot of M-polynomial of silicate network is shown in Figure 2.

Conclusion
In the paper, we gave some new formulas to find the vertex degree-dependent topological indices via M-polynomial. ese formulas were then applied to the silicate network. ese indices have valuable information about the molecular structure. e proposed new technique gave vast application in QSAR/QSPR to analyze the chemical structure. e results obtained are also plotted.

Data Availability
No data were used to support the study.

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