Some Degree-Based Topological Indices of Base-3 Sierpiński Graphs

In this paper, a recursive relation between base-3 Sierpiński graphs rank n and n-1 of some topostructural indices is studied. Based on this relation, The formulae of the First Zagreb index, Second Zagreb index, Randić connectivity index, sum-connectivity index, Geometric-Arithmetic index and Atom-Bond Connectivity index of base-3 Sierpiński graphs are derived.


Introduction
In this paper, we only consider undirected graph without loops and multiple edges. Let G be a graph, we denote ( ) V G and ( ) E G the vertex set and edge set of G, respectively. We denote by d v the degree of a vertex v of a graph G. We denote by In chemical graph theory, the vertices of the graph correspond to the atoms of molecules and the edges correspond to chemical bonds. Klavžar et al. [9] defined the Sierpiński graphs S (n,k) in 1997. The Sierpiński graphs S (n,k) were related to the Tower of Hanoi problems and the universal topological spaces problems. The authors defined S (n,k) as follows.
( ( , )) {1, 2, 3, , } , 1, 1 n V S n k k k n = ≥ ≥ ⋯ and two different vertices 1 2 ( , , , ) Hinz et al. [5] discussed the definition and properties of Sierpiński graph, Sierpiński-type graphs, Sierpiński triangle graphs and Sierpiński-like graphs. Because we only focus on Base-3 Sierpiński Graphs S (n,3), we simplify the notation from S (n,3) to S n . In Figure 1 the Base-3 Sierpiński Graphs are illustrated with n ∈ {1,2,3,4}. S 1 is K 3 and three vertices are labeled with 0, 1, and 2. S 2 consists of three copies of S 1 , if we consider a S 1 as a vertex, S 2 can be considered as another type of K 3 . The vertices labels of S 2 can be considered by adding S 1 vertex labels before to three copies of S 1 vertex labels. So S n+1 is composed with three copies of S n , S n+1 is another type of K 3 that all three vertices are replaced by S n , and S n vertex labels are composed by adding S 1 vertex labels before to three copies of S n vertex labels. The relationship of S n+1 and S n is illustrated in Figure  2. By the definition of the Base-3 Sierpiński Graphs S i , we have Observation 1. If G is an S n graph with n ≥ 2, then In chemical graph theory, the atoms of molecules can be considered as the vertices of graph and chemical bonds be considered as the edges of graph [12,14]. The mathematical quantities derived from a graph representation of the molecule are called graph invariants. Single indices derived from a molecular graph are called topological indices.
Topological indices are sensitive to one or more structural features of the molecule such as size, shape, symmetry, branching, and topological indices are divided into two categories: topostructural and topochemical indices [11]. In this paper, we only focus on some of the topostructural indices, including First Zagreb index 1 The First Zagreb index were introduced by Gutman and Trinajstić [6]. When they studied the molecular structure's π electron energy, they found the approximate expressions of M1 and M2. The First Zagreb index and the Second Zagreb index are very important topostructural indices in chemical graph theory [7].
The First Zagreb index 1 ( ) Zg G is defined as the sum of squares of the vertex degrees d u and d v of vertices u and v in G (see [6,7,13] The Second Zagreb index 2 ( ) Zg G is defined as the sum of squares of the vertex degrees d u and d v of vertices u and v in G (see [6,7,13]) and deduced as formula (2).
According to the above Zagreb indices, the First Zagreb polynomial 1 ( , ) Zg G x and the Second Zagreb polynomial 1 ( , ) Zg G x have been defined. They are defined as Kinkar and Gutman studied the properties of 1 ( , ) Zg G x , 2 ( , ) Zg G x for some chemical structures in 2004 [1].
The Randić index ( ) R G , proposed by Randić [10,12] in 1975, is well correlated with a variety of physico-chemical properties of alkanes. And this index has become one of the most popular molecular descriptors. The Randić index ( ) R G is defined on the ground of vertex degrees If more information about the Randić index of polymeric networks modelled by generalized Sierpiński graphs is wanted, please refer to Estradamoreno et al. [18].
The sum-connectivity index was introduced by Zhou and Trinajstić [15] which is very similar to the Randić connectivity index. The sum-connectivity index is defined (see [15]) as As molecular structure descriptor based on graph theory, ( ) X G and R(G) are both well related with each other and with the π-electronic energy of benzenoid hydrocarbons [16].
Fath-Tabar et al. were introduced the Geometric-Arithmetic index in 2010 (see [17]) is defined as The GA index is well correlated with a variety of physicochemical properties such as entropy, enthalpy of vaporization, standard enthalpy of vaporization, enthalpy of formation and acentric factor [3,8].
By Estrada et al. [4], the Atom-Bond Connectivity index is defined as The ABC index correlates well with the experimental heats of formation of alkanes [4]. The mathematical properties of this index have been studied extensively.
The Forgotten index ( ) F G is defined as the sum of cubic of the vertex degrees in G (see [13]) The Harmonic index H(G) (see [19]) is defined as More recently, motivated by the previous research on topological descriptors and their applications, Hosamani [20] proposed the Sanskruti index which can be utilized to guess the bioactivity of chemical compounds and shows good correlation with entropy of an octane isomers.
The Sankruti index of a graph G is defined as Darafsheh [2] have discussed Wiener index formulae, Szeged index formulae and PI-index formulae of intersection graph and Hypercube (n-Cube) graph by using automorphism group method, and asserted recursion process method be good at to the graph with the same shape repeated several times, so he gave Wiener index formulae of Linear Chain, Polyphenylene and Linear Phenylene graph by using recursion formulae method. The recursion formulae method will be used to derived the formulae of the First Zagreb index 1 ( ) Zg G , Second Zagreb index 2 ( ) Zg G , the first Zagreb polynomial, the second Zagreb polynomial, sum-connectivity index ( ) X G , Geometric-Arithmetic index ( ) GA G , Atom-Bond Connectivity index ABC(G), forgotten index F(G), harmonic index H(G) and Sankruti index ( ) G S of base-3 Sierpiński graphs.

Main Results
Let ϕ (x,y) be a function on R × → ℤ ℤ and a degree-based topological index f(G) is defined as By observing the structure of the base-3 Sierpiński graphs, we have the following recursive formulation of f(S n ): If n ≥ 3, then Proof. It can be verified that Zg 2 (S 1 ) = 12 and Zg 2 (S 2 ) = 66 thus the theorem holds for 1, 2 n = . Let ϕ(x,y) = x + y. Then the degree-based topological index f(G) in formula (1) is the first Zagreb index Zg 1 (G). By formula (1), we have If n ≥ 3, then Proof. It can be verified that Zg 2 (S 1 ) = 12 and Zg 2 (S 2 ) = 90 thus the theorem holds for 1, 2 n = . Now we consider the case n ≥ 3. Let ϕ(x,y) = xy. Then the degree-based topological index f(G) in formula (2) is the second Zagreb index Zg 2 (G). By formula (2), we have if n ≥ 3, then  Proof. It can be verified that   (7) is Geometric-Arithmetic index ( ) GA G . By formula (7), we have if n ≥ 3, then Now we consider the case n ≥ 3. Let 2 ( , ) x y x y xy φ + − = .
Then the degree-based topological index f(G) in formula (8) is ABC index ABC(G). By formula (8)   we consider the case 3 n ≥ . We know that Sankruti index is defined based on ( ) G s u which is the summation of the degrees of all the neighbors of a vertex u , and by formula (11), we give the recursive formulation as follows which is different as formula (12).