In fact, plotting L[-1/2, λ] against λ using a logarithmic scale (refer to Figure .1), we observe a curve consistent with the universal curve depicted in Figure (b) of . This alignment enables the validation of the methodology proposed for predicting the classic Randić index's value. Specifically, it delineates three distinct regimes: the predominantly isolated vertices regime (λ < 0.01), the transitional regime (0.01 < λ < 10), and the regime characterized by nearly complete graphs (λ > 10). When α = -1, R -1 is known as the second modified Zagreb index; which is related to the eigenvalues of the normalized Laplacian matrix of the graph . Thus, limit theorems concerning the second modified Zagreb index can be obtained from Theorems 2.1 and 2.3 by setting α = -1.