Multi-level distance labelings for generalized gear graphs

The emph{radio number} of $G$, $rn(G)$, is the minimumpossible span. Let $d(u,v)$ denote the emph{distance} between twodistinct vertices of a connected graph $G$ and $diam(G)$ be theemph{diameter} of $G$. A emph{radio labeling} $f$ of $G$ is anassignment 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 rangeof the labeling is its span. In this paper we show that$rn(J_{t,n})geqleft{ egin{array}{ll} frac{1}{2}(nt^{2}+2nt+2n+4), & hbox{when t is even;} frac{1}{2}(nt^{2}+4nt+3n+4), & hbox{when t is odd.} end{array} ight.$ oindent Further the exact value for the radio number of $J_{2,n}$ is calculated.