Some topological properties of uniform subdivision of Sierpiński graphs

: Sierpiński graphs are family of fractal nature graphs having applications in mathematics of Tower of Hanoi, topology, computer science, and many more diverse areas of science and technology. This family of graphs can be generated by taking certain number of copies of the same basic graph. A topological index is the number which shows some basic properties of the chemical structures. This article deals with degree based topological indices of uniform subdivision of the generalized Sierpiński graphs n S . The closed formulae for the computation of different kinds of Zagreb indices, multiple Zagreb indices, reduced Zagreb indices, augmented Zagreb indices, Narumi-Katayama index, forgotten index, and Zagreb polynomials have been presented for the family of graphs.

information on various properties of organic substances that depend upon their molecular structure. For this purpose, numerous topological indices were found and studied in the chemical literature (Todeschini and Cansonni, 2008). They used two zeroth order and two firstorder connectivity indices for the first time as descriptors in structure-property correlations in an optimization study. A set of new formulas for heat capacity, glass transition temperature, refractive index, cohesive energy, and dielectric constant were introduced -they were based on these descriptors. The Randić index has been used to parallel the boiling point and Kovats constants and was closely correlated with many chemical properties. A graph invariant that correlates the physico-chemical properties of a molecular graph with a number is called a topological index (Hansch and Leo, 1996). The first topological index was introduced by Wiener, a chemist, in 1947 to calculate the boiling points of paraffins in Wiener (1947). Zagreb indices, derived by Gutman and Trinajstić (1972), are used to study molecules and complexity of selected classes of molecules. Zagreb indices have found an interesting use in the QSPR/QSAR modeling and are useful in the study of anti-inflammatory activities of certain chemical instances.
A graph ( , ) S n G is known as the generalized Sierpiński graph of G of dimension n having vertex set {1, } , n l … and the set { , } t s defines an edge iff {1, , } i n ∃ ∈ … such as that:

s and t s E G ≠ ∈
• j i j i t s and s t = = if . j i > Sierpiński graphs appear in many fields of mathematics and other branches of science (Beaudou et al., 2010). One of the most necessary sorts of Sierpiński graphs are the Sierpiński gasket graphs. The Sierpiński gasket graph is outlined as ( , ) S n k with vertex set (1, 2, 3 , ) n … and there is an edge between 2 vertices The finite number of iterations in Sierpiński graphs gives Sierpiński gasket graphs and it is denoted by n S . n S is consisted of three linked copies of 1 n S − that are up, down left, and downright elements of n S . These graphs have been given by Scorer et al. (1944). These graphs play a vital role in dynamics system and probability and additionally as in psychological sciences. The generalized Sierpiński graph, ( , ) S n G , is constructed by repeating | | G times ( 1, ) S n G − and adding one edge between replica i and replica j of ( 1, ) S n G − wherever pq is an edge of G. In this paper we will discuss the uniform subdivision of n S and ( , ) S n G . We will denote subdivision of n S by ( ) n SD S and subdivision of ( , ) S n G by ( ( , )) SD S n G . We study three types of Sierpiński graphs and now we will calculate the degree based topological indices for these types.
The first and second Zagreb indices which are also known as the graph invariants are the first vertex degree based structures descriptors defined by Gutman and Trinajstić (1972), and elaborated in Gutman et al. (1975) These topological formula were first appeared in the in π energy of conjugated molecules. After ten years, Balaban et al. (1983) included: Here 1 ( ) M G and 2 ( ) M G are known as the first and second Zagreb index. Gutman, Xu, and Das (Gutman and Das, 2004;Xu and Das, 2012) found the use Zagreb indices in QSPR and in QSAR. These topological indices have been used to study "molecular complexness", "chirality, ZE-isomorphism", and Hetro-system. Their chemical applications and mathematical characteristics were studied by Das et al. (2013), Furtula et al. (2010), Gutman (2013, 2014), Nikolic et al. (2003), and Zhou (2004. Recently, Furtula and Gutman (2015) have found the use of this term. They suggested that since this topological index was forgotten so it should be named as forgotten index or simply F index it is defined as: Furtula and Gutman (2015) proved that the linear combination of 1 M F λ + has very accurate mathematical model of some physical properties of alkanes. Caporossi et al. (2010) introduced another index that is called a reduced second Zagreb index. Also this index has very important applications. It is defined as: Furtula et al. (2010) introduced the augmented Zagreb index in 2010. It is given as: Furtula et al. (2010) stated that it is a very important index for the study of the heat of formation in octanes and heptanes. If the exponent in AZI is replaces by -0.5 then it will become atom bound connectivity index. The results indicate that the AZI index gives the better results as compare to the ABC index. Shirdel et al. (2013) gave the third Zagreb index, which is defined as: This index is combination of F index and second Zagreb index, that is: By using the above relation, we study the F index with the Zagreb index. Narumi and Katayama (1984) defined the degree Ghorbani and Azimi (2012) introduced the first and second multiple Zagreb indices, which are defines as: The first multiple Zagreb index is square of Narumi-Katayama index. Fathtabar (2009) defined the first and second Zagreb polynomial of the graph G: where y is an attribute. In the following section some degree based topological indices for uniform subdivision of Sierpiński like graphs are studied.

Zagreb indices and Zagreb polynomials for uniform subdivisions of Sierpiński gasket graphs SD(S n )
The graphs of uniform subdivision of Sierpiński gasket for 1 3 n ≤ ≤ is given in Figure 1 and the size of ( ) n SD S is 3 n . There are 2 types of edges corresponding to the end vertices for 1 n > . The partition of edge set of ( ) n SD S is given in Table 3 In the vertex set there are 2 types of the vertices in ( ) V G according to the degrees. In the Table 2 vertex partition of vertex set of ( ) n SD S is given below.

Theorem 1
The first and second Zagreb indices for n G SD(S ) = is:

Theorem 2
The reduced second Zagreb index for n G SD(S ) = is: Table 1 we have:

Theorem 3
The third Zagreb index for n G SD(S ) = is:   Table 1 we have:

Theorem 5
The augmented Zagreb index for n G SD(S ) = is:

Zagreb indices and Zagreb polynomials for uniform subdivisions of Sierpiński gasket graphs for SD(S(n,C ))
The generalized Sierpiński graphs for the 3 3 3 ( (1, )), ( (2, )), and SD( (3, )) SD S C SD S C S C 3 3 3 ( (1, )), ( (2, )), and SD( (3, )) SD S C SD S C S C are given in the Figure 2. The size of 3 ( ( , )) SD S n C is 3(3 1). n − There are 2 types of edges corresponding to the degrees of end vertices for 1 n > . The edge partition for the set of 3 ( ( , )) SD S n C is given in the Table 3.
The order of the 3 ( ( , )) SD S n C is 3 (3 1) 3 2 n n − + . In the vertex set there are 2 types of the vertices in V(G) according to their degrees. The Table 4 illustrates a vertex partition of the ( ) V G of 3 ( ( , )) SD S n C . The following theorems represents the formulas for Zagreb indices and Zagreb polynomials for 3 ( ( , )) SD S n C .  (S(1,C 3 )), SD(S(2,C 3 )), and SD(S(3,C 3 )).      Table 3 we have:

Theorem 12
The third Zagreb index for 3 SD(S(n,C )) is:   Table 3 we have:

Theorem 15
The first and second multiple Zagreb indices for Here using the relation 2 ( ) PM G and the Table 3

SD S C SD S C
The order of the 4 ( , ) S n C is 4 (4 1) 4 3 n n − + . In the vertex set there are 2 types of the vertices in V(G) according to the degrees. The Table 4 represents vertex partition of V(G) of 4 ( ( , )) SD S n C .
The following theorems represent the formulas for Zagreb indices and Zagreb polynomials for 4 ( ( , )).

Theorem 19
The first and second Zagreb indices for 4 SD(S(n,C )) are:    Table 5 we have:

Conclusion
Topological indices are often studied with the help of their descriptors. In this paper certain degree based topological indices namely Zagreb indices, multiple Zagreb indices, reduced Zagreb indices, augmented Zagreb indices, Narumi-Katayama index, forgotten  ( ( , )) SD S n C . In future we will pay attention by subdividing the other families of Sierpiński graphs.