Radio number for middle graph of paths

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 ϕ:V(G)→{0,1,2,…} such that |ϕ(u)−ϕ(v)|≥diam(G)+1−d(u,v) for every pair of distinct vertices u, v of G. The span of ϕ is defined as span(ϕ)=max⁡{|ϕ(u)−ϕ(v)|:u,v∈V(G)}. The radio number rn(G) of G is defined as rn(G)=min⁡{span(ϕ):ϕ is a radio labeling of G}. In this paper, we determine the radio number for middle graph of paths.