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Multi-level distance labelings for generalized gear graphs
Authors: M. T. Rahim, M. Farooq, M. Ali, S. Jan
Number of views: 273
The \emph{radio number} of $G$, $rn(G)$, is the minimum
possible span. Let $d(u,v)$ denote the \emph{distance} between two
distinct vertices of a connected graph $G$ and $diam(G)$ be the
\emph{diameter} of $G$. A \emph{radio labeling} $f$ of $G$ is an
assignment of positive integers to the vertices of $G$ satisfying
$d(u,v)+|f(u)-f(v)|\geq diam(G)+1$. The largest integer in the range
of the labeling is its span. In this paper we show that
$rn(J_{t,n})\geq\left\{
\begin{array}{ll}
\frac{1}{2}(nt^{2}+2nt+2n+4), & \hbox{when t is even;} \\
\frac{1}{2}(nt^{2}+4nt+3n+4), & \hbox{when t is odd.}
\end{array}
\right.$\\
\noindent Further the exact value for the radio number of $J_{2,n}$ is calculated.