Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess