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

In this work we perform computational and analytical studies of the Randi\'c index $R(G)$ in Erd\"os-R\'{e}nyi models $G(n,p)$ characterized by $n$ vertices connected independently with probability $p \in (0,1)$. First, from a detailed scaling 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 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 Erd\"os-R\'{e}nyi models.