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 $\left\langle \overline{R}(G) \right\rangle = \left\langle R(G)\right\rangle/(n/2)$scaleswiththeproduct$\xi\approx np$, so we can define three regimes: a regime of mostly isolated vertices when $\xi < 0.01$($R(G)\approx 0$),atransitionregimefor$0.01 < \xi < 10$(where$0<R(G)< n/2$),andaregimeofalmostcompletegraphsfor$\xi > 10$($R(G)\approx n/2$)\mathrm{.}Then,motivatedbythescalingof$\left\langle \overline{R}(G) \right\rangle$, 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.