Journal of Advanced Research in Computer Technology & Software Applications Volume 1, Issue 1 - 2017, Pg. No. 46-50 Peer Reviewed Journal Article The Fractal of Circulant Graphs R Sangeetha1, G Jayalalitha2 1Research Scholar, Department of Mathematics, Bharathiar University, Coimbatore. 2Associate Professor, Department of Mathematics, R.V.S. College of Engineering and Technology, Dindigul 625002.

Abstract In this paper, we explore the clique polynomial of Circulant graphs. The lexicographic product of a Circulant graph of order n with S as the generating set is being computed (k times) and the closure of the union of the roots of reduced clique polynomial of powers is plotted and the graph approaches a fractal like object, the Julia set. We also try to explore the fractals of the clique polynomial of trees. Keywords: Circulant Graph, Fractal, Lexicographic Product, Clique Number, Clique Polynomial

Introduction lexicographic product power nears infinity. It is observed Graph polynomials are an important topic of interest as that the closure of these roots approaches fractals (Julia its roots represents a great deal of information. Given an set). arbitrary graph G on n vertices, we can compute a graph Basic Definitions polynomial by enumerating the number of occurrences of Definition 2.1 a particular property. There are several graph polynomials such as Chromatic polynomials, Tutte polynomials, matching Given a set ,the Circulant graph Cn,s is polynomials and Clique polynomials which have their own the graph with vertex set , and edge set E(G) = active areas of research. The roots of the characteristic where = the circular polynomial of a molecular graph are interpreted as distance modulo n is. energies of the electronic levels of the corresponding Example molecules [9].There may not be a direct application for The above graph is the Circulant graph C where the other graph polynomials to scientific fields; however it is 4,S natural to investigate the nature and location of the roots generating set S = {1} of these graph polynomials[2-5]. The roots of independence polynomials have been studied extensively[6-8] and important theories have been developed. Here we investigate the clique polynomial of a particular family of graphs called Circulant graphs. Various papers have been written on the theory of Circulant graphs. [4, 10-12] The Julia set consists of values such that arbitrarily small changes can cause drastic changes in the sequence of iterated function values and hence the behavior of the function on Julia set is chaotic. This paper explores the clique polynomial of Circulant graphs and its association to fractals - the Julia set.[1-3]­ As we know, the clique polynomial of a graph is the independence Definition 2.2 polynomial of its complement and hence the characteristics The independence polynomial [2] of G is the generating of graph composition of independence polynomial.[2] is polynomial for the sequence, where applied to clique polynomial and the roots of the polynomial is the largest for which, the independence number of G. is found using MATLAB code. These roots of polynomial Definition 2.3 are being characterized using the association property of the lexicographic product (graph composition) when the The clique number of a graph G denoted ω(G) by is the

Correspondence Author: R Sangeetha, Department of Mathematics, Bharathiar University, Coimbatore. E-mail Id: [email protected] How to cite this article: Sangeetha R, Jayalalitha G. The Clique Fractal of Circulant Graphs. J Adv Res Comp Tech Soft Appl 2017; 1(1): 46-50.

Copyright (c) 2017 Journal of Advanced Research in Computer Technology & Software Applications Sangeetha R et al. 47 J. Adv. Res. Comp. Tech. Soft. Appl. 2017; 1(1) number of vertices in a maximum clique of G. Equivalently; Example it is the size of a largest clique or maximal clique of G. Definition 2.6 Example A geometric pattern that is repeated at ever smaller scales The number of vertices in the maximum clique is 2 since to produce irregular shapes and surfaces that cannot be it is a triangle free graph. Hence the clique number of the Peterson graph ω(G) = 2 Definition 2.4

represented by classical geometry. Fractals are geometric The clique polynomial C (x) for the graph G is defined as G shapes that are self similar and have infinite copies of itself. the polynomial , where is the clique number of G, the coefficient of xk for k > 0 is the number Example of cliques ck in a graph with k vertices and the constant Theorem 2.1 term is 1. The independence polynomial of G[H] is given by Example

Julia Set In the above graph . The clique polynomial is given by, Proof By definition, the polynomial is given by,

Here C1 = 4 (no. of cliques in 1 vertex) and C2 = 4 (no. of (1) cliques in 2 vertices) where the number of independent is sets of cardinality k Definition 2.5 in G and is the number of independent sets of cardinality

For two graphs G and H, let G [H] be the graph with vertex j in H. Now, an independent set in G[H] of cardinality arises I by choosing an independent set in G of cardinality k for set V(G) X V(H) and such that the vertex (a, x) is adjacent to some and then, within each associated vertex (b, y) if and only if is adjacent to b (in G) or a = b and copy of H in G[H], choosing a non empty independent set x is adjacent to y (in H). The graph G[H] is the lexicographic in H, in such a way that the total number of vertices chosen product or composition of G and H. Thus the number of is l . But the number of ways of choosing this independent vertices and edges of the graph G[H] is given by V(G)V(H) set is exactly the coefficient of x1 in the equation (1). and E(H)V(G) + E(G)V(H) 2 where E(G) and E(H) is the edge set of the graph G and H respectively. Thus the independence polynomial of is given by, Sangeetha R et al. J. Adv. Res. Comp. Tech. Soft. Appl. 2017; 1(1) 48

iG[H] (x) = iG(iH(x) - 1) Hence the result has been proved. Hence the proof holds. Clique polynomial of triangle free graphs Corollary 2.1.1 Consider a graph that has no triangles. Then the maximum

clique that can be formed in it is K2 . Thus the general form The clique polynomial of is given by CG[H] (x) = CG(CH(x) - 1) of the clique polynomial of triangle free graph is of the form, Proof Since we already know the result, , the above theorem holds for clique polynomial and thus CG[H] (x) = Clique Fractal of Trees C (C (x) - 1) G H . Hence the result holds. Consider the clique polynomial of a tree and the roots of Fractals of Clique Polynomials the clique polynomial of the tree when raised to higher Basic Notations lexicographic powers are found. These roots when plotted approaches fractal. Here T is a tree of eight vertices and is the reduced seven edges. The clique polynomial of T approaches fractal, clique polynomial of G. the Julia set at its fourth power of lexicographic product. J(f) is the Julia sets of polynomials where is a polynomial The clique polynomial of the tree T is given by, of degree n. F(G) is the clique fractal of a graph G. Characterization of (Here no. of vertices = 8, no. of edges = 7) The lexicographic product (Graph Composition) satisfies Fig 3.2 and Fig 3.3 are obtained from the tree with the the associative property and hence it can be written as, 2 Gk = G[G[G...]] of a graph G. clique polynomial 1+ 8x + 7x which has been raised to K the lexicographic product power of 4 and power of 3 By this associability of graph composition we can write, respectively using MATLAB code. Fig 3.2 approaches the k k-1 k fractal object – The Julia set. G = G [G] and hence the corollary 2.1.1, implies, fG = f k-1o f k this in turn implies Roots (f k) = f o(-1)(Roots (f k-1)) G G G G G Clique Fractals of Circulant Graphs We have, Roots (f ) = f o(-1)(0) G G Definition 4.1 k o(-k) Hence, Roots (fG ) = fG (0) for each k > 1 (2) A walk in G is a finite non – null sequence W =v 0e1v1e2v2.....

Therefore. ekvk whose terms are alternately vertices and edges such (3) that for 1 < i < k, the ends of ei are vi-1 and vi Theorem 3.1 Definition 4.2 The clique polynomial F(G) of a graph G is precisely the A walk in which all the vertices are distinct is said to be Julia set J(fG) of its clique polynomial fG (x). Equivalently a path. F(G) is the closure of the union of the reduced clique roots of powers Gk, k = 1,2,...ꚙ Proof If G has clique number 1, then G is a null graph with n > 1 vertices and no edges. The reduced clique polynomial k fG (x) is given by, fG (x) = nx whose Julia set is {0}. Now G = k k k Kn for all k, and fG (x) = n x, whose set of roots is {0}. The union and limiting root set is J(fG) = {0}. Hence it is the clique fractal F(G) of graph G. Figure 3.1 Tree T If G has clique number at least 2, then f (x) has degree G Consider the path graph (P ) of two vertices which is at least 2. Since , we have f (0) = 0 and 2 G the same as the Circulant graph C where S = {1} is the . Thus 0 is a repelling fixed point off 2,{1} G generating set. (x) and therefore lies in J(fG(x)) . In particular, z0= 0 satisfies Since it is a triangle free graph, the clique polynomial of the hypothesis = J(f) for any z0ϵ J(f).hence, C graph is given by, (4) 2,{1} C( P ;x) = 1+ 2x + x2 The left hand side of the equation 4 is F(G)(from the 2 equation 3) Now the lexicographic product of this polynomial rose to higher powers and the union of its roots has been plotted[5] Thus F(G) = and it approaches the fractal object, Julia set. Below graphs Sangeetha R et al. 49 J. Adv. Res. Comp. Tech. Soft. Appl. 2017; 1(1)

Figure 3.2 F(T) (power 4)

Figure 4.2 F(P_2)(power 3)

Figure 3.2 F(T) (power 4) are obtained by plotting the roots of the lexicographic product of the clique polynomial of the Circulant graph

C2,{1} that is raised to the powers 2, 3 and 4. Fig 4.1, Fig 4.2 and Fig 4.3 are obtained from the clique Figure 4.3 F(P_2)(power 2) polynomial of the Circulant graph C with the generating set 2 Conclusion {1}. (P2) which has been raised to the lexicographic product The clique polynomial of trees and Circulant graphs has been explored. Clique polynomials of graphs are closed under lexicographic product. (Graph Composition) For higher products of a graph their clique polynomials approaches power of 4,3 and 2 respectively using MATLAB code. For to the Julia set. Thus, the clique polynomial of the Circulant the power 2 (Fig 4.3), we obtain the extreme four points graph is associated with a fractal. Further studies can be of the Julia set (Refer example of definition 2.6). For the done on the properties of the clique fractal compared power 3 (Fig 4.2), we obtain a circle shape which is a part to its associated Circulant graphs. Also polynomials such of the Julia set and for the power 4 (Fig 4.1), we observe as chromatic and rook polynomial can be explored in the graph approaching the Julia set – a fractal object and association with a fractal. thus we conclude the roots of the lexicographic product of the clique polynomial raised to higher powers associates References the fractal object. Here the considered case approaches 1. Barnsley M, Fractals Everywhere. San Diego: Academic the Julia set. Press. 1988. 2. Brown JI, Hickman CA, Nowakowski RJ. The independence fractal of a graph. Journal of Combinatorial Theory 2003; 87(2): 209-230. 3. Weisstein EW. Julia Set. Math World 1999 2005 [cited 2005 11/20/05]; Available from: http://mathworld. wolfram.com/JuliaSet.html. 4. Boesch F, Tindell R. Circulants and their Connectivities. Journal of 1984; 8(4): 487-499. 5. Brown JI, Nowakowski RJ. Bounding the roots of independence polynomials, Ars Combin 2001; 58: 113–120. Figure 4.1 F(P_2) (power 4) Sangeetha R et al. J. Adv. Res. Comp. Tech. Soft. Appl. 2017; 1(1) 50

6. Brown JI, Dilcher K, Nowakowski RJ. Roots of Independence Polynomials of Well-Covered Graphs, Journal of Algebraic Combinatorics 2000; 11(3): 197- 210. 7. Brown JI, Hickman CA, Nowakowski RJ. On the Location of Roots of Independence Polynomials. Journal of Algebraic Combinatorics 2004; 19: 273-282. 8. Brown JI, Nowakowski RJ, Bounding the Roots of Independence Polynomials, Ars Combinatoria 2001; 58: 113-120. 9. Levit VE, Mandrescu E, The Independence Polynomial of a Graph { A Survey, submitted (preprint). 10. Adam A, Research Problem 2-10, Journal of Combinatorial Theory 1967; 2: 309. 11. Alspach B, Parsons T, Isomorphism of Circulant Graphs and Digraphs, Discrete Mathematics 1979; 25: 97-108. 12. Wu J, Song Z, A Class of Star Extremal Circulant Graphs. Journal of Southeast University (English Edition) 2002; 18: 177-179.

Date of Submission: 2017-04-14 Date of Acceptance: 2017-04-28