Multi-level distance labelings for generalized gear graphs

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 $rn(J_{t,n})geqleft{ begin{array}{ll} frac{1}{2}(nt^{2}+2nt+2n+4), & hbox{when t is even;} \ frac{1}{2}(nt^{2}+4nt+3n+4), & hbox{when t is odd.} end{array} right.$\ noindent Further the exact value for the radio number of $J_{2,n}$ is calculated.