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In this work we perform computational and analytical studies of the Randi\'cindex $R(G)$ in Erd\"os-R\'{e}nyi models $G(n,p)$ characterized by $n$ verticesconnected independently with probability $p \in (0,1)$. First, from a detailedscaling analysis, we show that $\left\langle \overline{R}(G) \right\rangle =\left\langle R(G)\right\rangle/(n/2)$ scales with the product $\xi\approx np$,so we can define three regimes: a regime of mostly isolated vertices when $\xi< 0.01$ ($R(G)\approx 0$), a transition regime for $0.01 < \xi < 10$ (where$0<R(G)< n/2$), and a regime of almost complete graphs for $\xi > 10$($R(G)\approx n/2$). Then, motivated by the scaling of $\left\langle\overline{R}(G) \right\rangle$, we analytically (i) obtain new relationsconnecting $R(G)$ with other topological indices and characterize graphs whichare extremal with respect to the relations obtained and (ii) apply theseresults in order to obtain inequalities on $R(G)$ for graphs inErd\"os-R\'{e}nyi models.

Computational and analytical studies of the Randi\'c index in Erd\"os-R\'{e}nyi models