Polynomials and General Degree-Based Topological Indices of Generalized Sierpinski Networks

College of Modern Service Industry, Hefei College of Finance and Economics, Hefei 230601, China Department of Mathematics, University of the Punjab, Lahore, Pakistan Department of Mathematics and Statistics, the University of Lahore, Lahore, Pakistan Department of Mathematics, Division of Science and Technology, University of Education Lahore, Lahore, Pakistan School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China


Introduction
Networks are studied in graph theory as spacial graphs having nice properties. A fractal is a geometric structure showing capabilities of self-similarity and repetition throughout the structure. Some commonly used fractalnatured networks appear useful in the study of complex systems present in both natural and artificial systems such as computer systems, music, brain, and social networks, allowing further development of the field in network science. ese networks are largely used to capture images of complex structures and predict behaviour of biological processes like nerve dendrites and growth pattern of bacteria in any culture. ese networks are very close to WK-recursive networks used in the design and implementation of local area networks as well as parallel processing systems. A special case of generalized Sierpinski graphs is the duality of Apollonian network, which displays the prominent scalefree small-world characteristics as observed in various real networks [1]. Because of diverse properties, these networks have been a subject matter of recent research.
A topological invariant is a mathematical property that remains invariant under the isomorphism of topological structures. In chemical graph theory, one is in search of the most general graph invariant that can capture at most all numerical descriptors of the structure. A new polynmial graph invariant, M-polynomial, was introduced in [2] by Deutsch and Klavžar. is invariant plays similar role in determining degree-based indices as played by Hosoya polynomial in determining distancebased indices.
Applications of the aforementioned polynomial can be traced in mathematical chemistry and pharmacology. Many degree-based topological indices can be derived using successive operations of differentiation and integration. e most interesting application of the M-polynomial is that almost all degree-based graph invariants, which are used to predict physical, chemical, and pharmacological properties of organic molecules, can be recovered from it (for more information, see [3][4][5][6][7][8][9]). e M-polynomial and related topological indices have been studied for several classes of graphs. In 2015, Deutsch and Klavžar gave M-polynomial, first Zagreb, and second Zagreb indices of polyomino chains, starlike trees, and triangulenes [2]. In 2017, Mobeen gave M-polynomial and several degree-based topological indices of titania nanotubes [10], of triangular boron nanotubes [11], and of Jahangir graph [12]. In 2017, Ajmal gave M-polynomial and topological indices of generalized prism network [13]. e authors computed M-polynomial and related indices of some benezoid systems in [14], hex-derived network in [15], and V-phenylenic nanotubes and nanotori [16]. Today, the computation of this polynomial is considered as one of the most important areas of research in the fields of molecular computing and degree-based indices.
Several degree-based topological indices, which play important role in mathematical chemistry, can be recovered from the M-polynomial.
is article is organized as follows. In Section 2, we give definitions of graph, topological indices, M-polynomial, and Sierpinski networks. e main results about the Sierpinski networks are given in Section 3; this section also covers the results about the topological indices.

Preliminary Notes
where V is the set of vertices and E is the set of edges. Any standard text on graph theory can be used as fundamental concepts related to graph and network theory.
e edge e between two vertices u and v is denoted by (u, v). e degree of a vertex u, denoted by d u , is the number of edges incident to it. A path from a vertex v to a vertex w is a sequence of vertices and edges that starts from v and stops at w. e number of edges in a path is the length of that path. A graph is said to be connected if there is a path between any two of its vertices.
A molecular graph is a representation of a chemical compound in terms of graph theory. Specifically, molecular graph is a graph whose vertices correspond to (carbon) atoms of the compound and whose edges correspond to chemical bonds [34]. For our computational models, we consider graphs which are hydrogen suppressed.
In the following, by G we shall mean a connected graph, E its edge set, V its vertex set, e � (u, v) its edge joining the vertices u and v, and d u degree of its vertex u.
e first Zagreb polynomial, second Zagreb polynomial, and forgotten polynomial of G are, respectively, defined as A function I which assigns to every connected graph G a unique number I(G) is called a graph invariant. Instead of the function I, it is custom to say the number I(G) as the invariant.
Definition 4 (see [34]). An invariant of a molecular graph which can be used to determine structure-property or structure-activity correlation is called the topological index. A topological index is an invariant of the graph.
A topological index is said to be degree-based if it depends on degrees of the vertices of the graph. e following are definitions of some degree-based indices that have connection with the M-polynomial.

Definition 5.
e first Zagreb, second Zagreb, and second modified Zagreb indices of G are defined, respectively, by

Definition 6.
e generalized Randić and reciprocal generalized Randić indices of G are defined, respectively, by

Definition 7.
e symmetric division index of G is defined by the relation Definition 8. e harmonic, inverse sum, augmented Zagreb, and forgotten indices are defined, respectively, by A remarkable property of the M-polynomial is that all the above degree-based indices can be recovered from it, using the relations given in Table 1.
Another approach to reach at many Zagreb indices is to compute (α, β)-Zagreb index introduced by Azhari et al. in [35]. It can be defined as 2 Complexity Sarkar et al. [36] computed Z α,β for some derived networks. e authors computed this index for V-phenylenic nanotubes and nanotori molecules [37] and circumcoronene series of benzenoid in [38]. is is somehow more general index, and several other indices can be calculated evaluating it at certain values. e following table establishes relation of other Zagreb indices as particular cases of this general Zagreb index. e redefined Zagreb index was first introduced in 2013 by Ranjini et al. [39] and is defined as to generalize the Zagreb indices. Table 2 gives mathematical derivations of these indices from (α, β)-Zagreb index [40]. e authors computed this index for some nanotubes and graphs in [41,42]. Table 2 relates some of Zagreb types indices with (α, β)-Zagreb index.
Definition 9. Let V t be a set of words of size t on the set of alphabets of V. e letters of word u of length t are denoted by u 1 , u 2 , . . . , u t . e concatenation of two words u and v is denoted by uv. e graph S(K n , t), introduced in [43 For n � 3, we receive Tower of Hanoi graphs. Later in [44] these graphs have been called as Sierpinski networks.
Sierpinski networks form an extensively studied class of graphs of fractal nature applicable in topology, mathematics of Tower of Hanoi, computer science, and in other disciplines. A large number of properties like physicochemical properties, thermodynamic properties, chemical activity, biological activity, and so on are determined by the chemical applications of graph theory. In [45], the authors computed molecular topological properties of Sierpinski networks and derived closed formulas for the atom-bond connectivity index, geometric-arithmetic index, and fourth and fifth version of these topological indices for Sierpinski networks. In [46], the authors discussed some matrical aspects of these networks. Recently, the authors computed various versions of degree-based topological indices of some variants of these networks using direct approach [47]. But connectivity polynomials such as the M-polynomial and generalized Zagreb Index have not been computed for this network. Connectivity and number of connected components of this network have been computed recently in [48]. Hamiltonicity of Sierpinski networks has been discussed as well in [48]. Recently in [49], the author used harmonic Sierpinski gasket as a geometric configuration of small antennas. In [50], the authors discussed some energies of the Sierpinski gasket. e authors in [51] computed strong metric dimension of generalized S n k having isolated pendant nodes. Total colorings of S n k were discussed in [52]. e authors in [53] discussed all kinds of shortest paths in Sierpinski networks with the help of newly designed algorithm. For comprehensive survey and classifications of S n k , we refer the readers to [54]. In this paper, we compute M-polynomial and some degreebased topological indices of Sierpinski graphs S n k . e Sierpinski network S n 3 has 3 n , n ≥ 2 vertices and 3(3 n − 1)/2 edges. Figure 1 shows S 2 3 where S 3 is triangle. Now, we present S 4 3 in Figure 2. e Sierpinski network S n 4 has 4 n , n ≥ 2 vertices and 4(4 n − 1)/3 edges. Figure 3 depicts first two instances of Sierpinski network S n 4 for n � 1 and n � 2. Note that S n 3 and S n 4 contain only vertices of degrees 2 and 3. Edge partition of S(n, k) Sierpinski network is based on degrees of end vertices of each edge.

Main Results
In this section, we give M-polynomial, related indices, and some general Zagreb indices of Sierpinski networks S n k .

Index Relation with the M-polynomial
Where

Index
Derivation from

Complexity 3
Proof. We prove it by finding sets that partition vertices and edges of S n k . e disjoint sets of vertices of S n k are e edge partition of S n k contains the sets  4 Complexity Proof. Before giving step-by-step proofs of all items, we find D x , D y , S x , and S y .
2k 2α x k y k , Now the indices are e harmonic, inverse sum, and augmented indices of S n k , n ≥ 1, are (1) H(S n k ) � ((4k)/(2k − 1)) + ((k n+1 − 5k)/(2k)) e first Zagreb polynomial, second Zagreb polynomial, forgotten polynomial, and forgotten index of Proof. To prove this, we again have to refer to Table 1 where edge partitions of S n k , n ≥ 1 are given. en, by using relation in (2), we have Remaining three parts of this theorem are just easy consequences of the result of Z α,β .

Conclusions and Discussion
In the present article, we computed some connectivity polynomials and many degree-based descriptors of Sierpinski networks specifically the closed form of M-polynomial and generalized Zagreb index. In this section, we give graphical representation of our results. where G is any graph. We here give short analysis of the indices depending upon n and k. For analysis, we use 3D graphs developed by GeoGebra. First 3D surface is plotted for forgotten index (see Figure 4). Slices along different axes represent the forgotten index for different values of n and k. Next Figure 5 is plotted for inverse sum index. It can be seen that the surface has some part as of a saddle where along one parameter its curvature is opposite to the other parameter. Figure 6 is the plotted surface for modified second Zagreb index of (S k n ). Here the surface is parabolic cylinder and curves on the surface show the curve for modified  second Zagreb index for different combinations of parameters n and k. ese curves are parabolic in nature which depict the behaviour of the respective index with respect to particular parameters involved. Similarly, Figure 7 represents the surface plotted for redefined Zagreb index of (S k n ). Different graphs shown on the surface represent critical behaviour of this index for the parameters n and k. It is trivially observed that for some combinations, it is a straight line, and for some, it is parabolic.
ese facts can be used as raw inputs in networking and fractals where these networks play a significant role. All surfaces for other computed indices can be easily plotted and interpreted similarly. Physicochemical properties of the molecules modelled on the S k n networks can be forecasted using the results given above.

Data Availability
No data were used to support this study.

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

Authors' Contributions
e manuscript was drafted and written by MMM and ZH. e concepts were initially developed by JBL, CF, and MA. All authors have read and approved the final manuscript. 8 Complexity