Entropy and Multi-Fractal Analysis in Complex Fractal Systems Using Graph Theory

Abstract

In 1997, Sierpinski graphs, S(n,k), were obtained by Klavzar and Milutinovic. The graph S(1,k) represents the complete graph Kk and S(n,3) is known as the graph of the Tower of Hanoi. Through generalizing the notion of a Sierpinski graph, a graph named a generalized Sierpinski graph, denoted by Sie(Λ,t), already exists in the literature. For every graph, numerous polynomials are being studied, such as chromatic polynomials, matching polynomials, independence polynomials, and the M-polynomial. For every polynomial there is an underlying geometrical object which extracts everything that is hidden in a polynomial of a common framework. Now, we describe the steps by which we complete our task. In the first step, we generate an M-polynomial for a generalized Sierpinski graph Sie(Λ,t). In the second step, we extract some degree-based indices of a generalized Sierpinski graph Sie(Λ,t) using the M-polynomial generated in step 1. In step 3, we generate the entropy of a generalized Sierpinski graph Sie(Λ,t) by using the Randić index.

Keywords

• M-polynomial
• Shannon’s entropy
• graph entropy
• Randić index
• fractals
• generalized Sierpinski graphs