Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Girth, minimum degree, independence, and broadcast independence   An independent broadcast on a connected graph $G$ is a function $f:V(G)\to
\mathbb{N}_0$ such that, for every vertex $x$ of $G$, the value $f(x)$ is at
most the eccentricity of $x$ in $G$, and $f(x)>0$ implies that $f(y)=0$ for
every vertex $y$ of $G$ within distance at most $f(x)$ from $x$. The broadcast
independence number $\alpha_b(G)$ of $G$ is the largest weight
$\sum\limits_{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$.
  It is known that $\alpha(G)\leq \alpha_b(G)\leq 4\alpha(G)$ for every
connected graph $G$, where $\alpha(G)$ is the independence number of $G$. If
$G$ has girth $g$ and minimum degree $\delta$, we show that $\alpha_b(G)\leq
2\alpha(G)$ provided that $g\geq 6$ and $\delta\geq 3$ or that $g\geq 4$ and
$\delta\geq 5$. Furthermore, we show that, for every positive integer $k$,
there is a connected graph $G$ of girth at least $k$ and minimum degree at
least $k$ such that $\alpha_b(G)\geq 2\left(1-\frac{1}{k}\right)\alpha(G)$. Our
results imply that lower bounds on the girth and the minimum degree of a
connected graph $G$ can lower the fraction $\frac{\alpha_b(G)}{\alpha(G)}$ from
$4$ below $2$, but not any further.

33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph