Limit theorems for the classic Randić connectivity index can be derived from Theorems 2.1 and 2.3 by taking α = -1/2. In the dense case, we observe that its asymptotic behavior is independent of the parameter p, i.e, its determination comes only from the structure; whereas the limits of its variations do fluctuate: as a function of p, the long-term behavior of the generalized Randić index is increasing for -1/2 < α and decreasing for α < -1/2. By the way, as a function of α, the long-term behavior of the generalized Randić index is decreasing for p ∈ (0, 1). Throughout this section we define L[α, λ] = λc 2 α,λ . Corollary 3.1. For both the dense and sparse case of ER graphs, the classic Randić index of G n,p is asymptotically concentrated around its mean. More precisely, as n → ∞,