Complexity Measures for Self-Similar and Chemical Graphs Using Entropies and Generalized Fractal Dimensions

Nonlinear analysis finds numerous applications in understanding the intricate characteristics of generalized Sierpinski graphs or chemical graphs. It aids in discerning the fractal patterns inherent in self-similar structures, deciphering the dynamic interactions within chemical bonds, and assessing the complexity of molecular arrangements. Furthermore, it plays a crucial role in unraveling the emergent behaviors of complex systems, offering insights applicable to diverse fields such as materials science, chemistry, biology, network theory, and beyond. In this paper, the complexity of generalized Sierpinski graphs and chemical graphs is analyzed using fractal-based nonlinear measures computed iteratively. Nonlinear measures assess intricate relationships within graphs by taking into account interactions that diverge from linear patterns. In particular, generalized Sierpinski graphs are derived from an increased number of iterative processes, which can also be viewed as a form of self-similar graph. The purpose of the study is to compare nonlinear measurements such as generalized fractal dimensions, Renyi entropy, and Tsallis entropy for extended Sierpinski graphs and benzenoid chemical graphs to understand the relationship among them. However, the benzenoid graph has self-similarity as a result of its chemical bonding structure. Furthermore, the paper assesses the efficacy of these methods in terms of iteration wise comparisons, employing a wide range of orders. The computational results in this paper indicate that the complexity of the representative graphs grows with the number of iterations, reflecting increased self-similarity as the values of fractal-based measures rise for both generalized Sierpinski graphs and benzenoid chemical graphs. This comparative research will enable us to quantify non-linearity and analyze self-similarity in real-time graphical networks or chemical structures.