Radio number for middle graph of paths

Abstract: For a connected graph $G$, let $diam(G)$ and $d(u,v)$ denote the diameter of $G$ and distance between $u$ and $v$ in $G$. A radio labeling of a graph $G$ is a mapping $\varphi : V(G) \rightarrow {0,1,2,...}$ such that $|\varphi(u)-\varphi(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$. The span of $\varphi$ is defined as span($\varphi$) = $\max{|\varphi(u)-\varphi(v)| : u, v \in V(G)}$. The radio number $rn(G)$ of $G$ is defined as $rn(G)$ = $\min{$span($\varphi$) : $\varphi$ is a radio labeling of $G}$. In this paper, we determine the radio number for middle graph of paths.