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In this work we perform computational and analytical studies of the Randić index R(G) in Erdös-Rényi models G(n, p) characterized by n vertices connected independently with probability p ∈ (0, 1). First, from a detailed scaling analysis, we show that R(G) = R(G) /(n/2) scales with the product ξ ≈ np, so we can define three regimes: a regime of mostly isolated vertices when ξ < 0.01 (R(G) ≈ 0), a transition regime for 0.01 < ξ < 10 (where 0 < R(G) < n/2), and a regime of almost complete graphs for ξ > 10 (R(G) ≈ n/2). Then, motivated by the scaling of R(G) , we analytically (i) obtain new relations connecting R(G) with other topological indices and characterize graphs which are extremal with respect to the relations obtained and (ii) apply these results in order to obtain inequalities on R(G) for graphs in Erdös-Rényi models.

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