Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp bounds for the Randic index of graphs with given minimum and
  maximum degree   The Randi{\' c} index of a graph $G$, written $R(G)$, is the sum of $\frac
1{\sqrt{d(u)d(v)}}$ over all edges $uv$ in $E(G)$. %let $R(G)=\sum_{uv \in
E(G)} \frac 1{\sqrt{d(u)d(v)}}$, which is called the Randi{\' c} index of it.
Let $d$ and $D$ be positive integers $d < D$. In this paper, we prove that if
$G$ is a graph with minimum degree $d$ and maximum degree $D$, then $R(G) \ge
\frac{\sqrt{dD}}{d+D}n$; equality holds only when $G$ is an $n$-vertex
$(d,D)$-biregular. Furthermore, we show that if $G$ is an $n$-vertex connected
graph with minimum degree $d$ and maximum degree $D$, then $R(G) \le \frac n2-
\sum_{i=d}^{D-1}\frac 12 \left( \frac 1{\sqrt{i}} - \frac
1{\sqrt{i+1}}\right)^2$; it is sharp for infinitely many $n$, and we
characterize when equality holds in the bound.

37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph