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A radio labeling of a simple connected graph G is a function c:V(G) \to Z_+ such that for every two distinct vertices u and v of Gdistance(u,v)+|c(u)-c(v)|\geq 1+ diameter(G).The radio number of a graph G is the smallest integer M for which there exists a labeling c with c(v)\leq M for all v\in V(G). The radio number of graphs of order n and diameter n-1, i.e., paths, was determined in [7]. Here we determine the radio numbers of all graphs of order n and diameter n-2.