### Radio labelings of graphs

A radio labeling of a connected graph $G$ is an assignment of
distinct positive integers to the vertices of $G$, with $x \in V(G)$
labeled $c(x)$, such that
$$d(u, v) + |c(u) - c(v)| \geq 1 + \diam G$$
for every two distinct vertices $u, v$ of $G$, where
$\diam G$ is the diameter of $G$. The radio number
$rn(c)$ of a radio labeling $c$ of $G$ is the maximum
label assigned to a vertex of $G$. The radio number
$rn(G)$ of $G$ is $\min\{rn(c)\}$ over all radio labelings $c$ of $G$.
The radio numbers of some well known graphs are studied.
It is shown that if $G$ is a connected graph of order $n$
and diameter $2$, then $n \leq rn(G) \leq 2n-2$
and that for every pair $k, n$ of integers
with $n \leq k \leq 2n-2$, there exists a connected
graph of order $n$ and diameter $2$ with $rn(G) = k$.
A characterization of connected graphs of order $n$
and diameter $2$ with prescribed radio number is presented.