Zagreb-type indices are topological indices derived from the degrees of nodes. The first Zagreb index, the F-index, and the Y-index represent the sum of the squares, cubes, and fourth powers of all node degrees, respectively. These indices are valuable for understanding the chemical reactions, physical characteristics, and biological activities of various substances. In this study, we explore the connection between Y-index and the graph Laplacian spectrum. Additionally, we introduce the fractal graphs based on star graphs, a class of extended Vicsek graphs, and derive the rules for eigenvalue evolution between two generations of the graph. Ultimately, we provide exact closed-form expressions for the first Zagreb index, F-index, and Y-index of the fractal graphs based on star graphs by using spectral graph theory.
Jia et al. (Wed,) studied this question.