INDEPENDENCE POLYNOMIAL OF THE SIERPINSKI GASKET GRAPH AND THE TOWER OF HANOI GRAPH

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In dynamical systems, the most well-known fractals are the Sierpinski gasket graph, also known as the Sierpinski fractal, and the Tower of Hanoi graph. In this paper, we investigate the nth generated independence polynomial of these graphs as well as its properties. In order to partition its spanning subgraphs, we use iterative patterns of the Sierpinski graph and the Hanoi graph. Furthermore, we consider the relationship between the Tower of Hanoi graph and the Sierpinski fractal in terms of their independence polynomial.


INTRODUCTION
The study of system which is moving or changing in time is called dynamics .Fractals are geometric objects that are static images which is self similar and whose dimension exceeds its topological dimension.The Sierpinski fractal, also known as the Sierpinski gasket graph, is a well-known fractal and is denoted by SGg.The Tower of Hanoi is a classic puzzle created in 1883 by Edward Lucas , and the Tower of Hanoi graph is denoted by T Hg.The graphs such as Tower of Hanoi and Sierpinski are came from these games like sources. As a result, it is interesting to investigate graphs whose drawings can be approximated to Sierpinski or Hanoi graphs.In the past few years, we have used these structures extensively in topology, algorithm analysis, discrete mathematical structures, and various types of networks [1].Both graphs have the same vertex set but can be labeled differently.This distinction allows for different approaches to both graphs.The relationship between these two graphs is useful in studying the classic TH task [2].
Determining the independence number and polynomial for a general graph is NP-hard [3], and its exact computation is extremely difficult.As a result, we choose these two graphs that are exactly tractable, and this is a common research path for NP-complete problems.The problem of determining independence polynomial has found widespread application in various of fields, most particularly theoretical computer science, coding theory, and wireless networks [4]. The independence polynomials are almost everywhere, but it is an NP complete problem to determine the independence polynomial of a graph.

SIERPINSKI TRIANGLE
The Sierpinski Gasket is a fascinating mathematical term that visualizes a variety of mathematical concepts. It was named after the Polish mathematician who suggested the concept first. The definition is a fractal, which is an equilateral triangle that can be subdivided into four different equilateral triangles, each 1 7074 K.U. SREEJA, P.B. VINODKUMAR, P.B. RAMKUMAR then be subdivided again, and the fractal continues in this manner. The mathematical concept of self-similarity is one of the most important ideas derived from the Sierpinski Gasket.
The SGg can be constructed recursively.Let S n represent the n th generation graph where n ≥ 1.If n = 1,S 1 is an equilateral triangle with three vertices and three edges.If n = 2, remove from the middle a triangle whose dimensions are exactly half that of the original triangle.This produces three smaller equilateral triangles, each with one-half the dimensions of the original triangle, generating S 2 .For n > 2,S n is obtained from S n−1 by performing the aforementioned operations on each triangle in S n−1 .The graphs obtained through n iterations of this method that leads to the Sierpinski gasket graphs and denoted by S n .The Sierpinski fractal is used in digital image steganography, dynamic systems, probability, and psychology [6].

THE TOWER OF HANOI
The Tower of Hanoi and its variations, which were first published in 1883 under the title N.
Claus and sold as a puzzle game, have been the subject of hundreds of scientific papers in fields ranging from mathematics to computer science to psychology [8].In fact, when we look at the set of all possible puzzle configurations as defined in Analyzing the Tower of Hanoi, as well as the relationships between those configurations, we find a surprising connection to the Sierpinski gasket.
T Hg can be constructed by the recursive modular method [9].For n = 0,H 0 is the complete graph K 3 .For n ≥ 1,H n is obtained from three copies of H n−1 joined by three new edges, each one connecting a pair of vertices from two different replicas of H n−1 and these three new edges are called special edges. Geometrically the Tower of Hanoi graph can be interpreted as the intersection graph of the remaining triangles after the n th iteration of S n .

INDEPENDENCE POLYNOMIAL OF SGg OF ORDER 1,2 AND 3
We can calculate independent sets of different orders of SGg by labeling the vertices differently.
For n = 2, S 2 contains the following independent sets of different orders.There is only one independent set of order three, and there are six independent sets of order two.Because all of the vertices of S 1 are order 1 independent sets, there are a total of six elements.The empty set is regarded as an independent set of order zero.Therefore independence polynomial of S 2 is given by z 3 + 6z 2 + 6z + 1.
For n = 3, S 3 contains independent sets of 2 elements of order 6,52 elements of order 5,138 elements of order 4,166 elements of order 3,78 elements of order 2,15 elements of order 1 and empty set of order 0.Therefore independence polynomial of S 3 is given by 2z 6 + 52z 5 + 138z 4 + The independence number of the SGg of order 2 is three, and the independence number of order 3 is six. Also, the maximum number of independent sets for S 2 and S 3 is one and two, respectively.
As a result, the independence polynomial of order 2 and 3 of SGg satisfies the following standard results.This demonstrates that the independence polynomial of our Sierpinski graph is unique.

THE INDEPENDENCE POLYNOMIAL OF THE SGg OF ORDER n
The class of self-similar groups contains many interesting properties. The Sierpinski graph and the Tower of Hanoi graph use this self-similar representation properties.We must analyze the relationship between spanning subgraphs for the independence polynomial of order n. The following observations are true.
The restrictions of special vertices determine whether or not the spanning subgraphs belong to the recursive formula for SGg. We will use the following partitions to find the number of independent sets on S n , which corresponds to the result obtained in [11].The graph's three outermost vertices of degree two play a crucial role in this partition.Furthermore, we should consider cases where the outermost vertices belong to the same connected components separately. We have the spanning subgraphs of S n using [11] as   The S n subgraphs can be represented graphically as shown in Fig.3. If we connect two outmost vertices in our diagram with a solid line, it means they are in the same connected component, according to our convention. From Fig.3, the subgraphs of S n can be written as S n =G 3,n ∪ G p 1,n ∪ G q 1,n ∪ G r 1,n ∪ G 0,n for each n ≥ 1.

INDEPENDENCE POLYNOMIAL OF HANOI GRAPH OF ORDER 1 AND 2
For n = 1,the independence polynomial of the T Hg is I(H 1 , z)=1 + 3z.
For n = 2,it is easy to check that the number of independence sets of order 3 is 18, of order 2 is 24, of order 1 is 9 and of order 0 is 1.Therefore independence polynomial of T Hg of order two is I(H 2 , z) = 18z 3 + 24z 2 + 9z + 1.
In comparison to the following corollary, the independence polynomial's uniqueness is established.
Corollary 6.1. [12] The number of vertices of H n is 3 n+1 and the number of edges is 3 n+2 −3 2 .

INDEPENDENCE POLYNOMIAL OF T Hg OF ORDER n
In this case, we can easily ignore the labeling of the vertices of the Tower of Hanoi graph and consider it as an unlabeled graph.
We will use the following partitions to find the number of independent sets on H n , which corresponds to the result obtained in [13]. We have the spanning subgraphs of H n using [13]as

NUMBER OF INDEPENDENT SETS ON H n
The number of independent sets will be calculated using the above partitions of the Hanoi graph.Let r n be the number of independent sets such that the vertex subset does not contain all three outmost vertices. Let s n be the number of independent sets in which the vertex subset contains only one of the three outmost vertices.Let t n be the number of independent sets in which the vertex subset contains the exact two specified vertices of the three outmost vertices.
Let u n be the number of independent sets such that the vertex subset contains all three outmost vertices.
Proposition 8.1. The number of independent sets of H n is given by i n =r n + s n + t n + u n .
For the general result,by utilizing the above outcome for each n ≥ 1, H n =M 3,n ∪ M p 1,n ∪ M q 1,n ∪ M r 1,n ∪M 0,n , we have the total number of independent sets is equal to the sum of the independent sets of all the above partitions. Hence the result.

THE RELATIONSHIP BETWEEN THE INDEPENDENCE POLYNOMIAL OF SGg AND THE INDEPENDENCE POLYNOMIAL OF T Hg
SGg of order 3 is analogus to T Hg of order 3.The only difference between the independence polynomials of SGg and T Hg is that the special edges do not determine a spanning subgraph of S n , and these edges do not belong to any of the copies of S n . However, in the case of the Hanoi graph, we must specify how many special edges belong to its spanning subgraph.
Proposition 9.1. [12] The number of independent sets of T Hg of order 3 is 108144.
Remark 9.2. The independence polynomial of SGg of order 3,S 3 is not same as the independence polynomial of T Hg of order 3. We have I(S 3 , z) = 2z 6 + 52z 5 + 138z 4 + 166z 3 + 78z 2 + 15z + 1 and it contains 452 independent sets.But T Hg of order 3 contains 108144 independent sets and is a polynomial of degree 12.So both the graph has different independence polynomials.

CONCLUSION
It is interesting to study the independence polynomial of a fractal network.In this paper,we investigate the independence polynomial of Sierpinski graph and Tower of Hanoi graph.First, we partition the independent sets of these graphs into five disjoint subsets, and then we inspect its various possible contributions.Finally, we derive the recursive formula for their independence polynomial based on their self-similar structure. Using these formulas, we can define the exact number of independent sets and the independence number of these graphs.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.