Radio Labeling

Let  be a finite, connected, undirected graph with graph diameter  and graph distance  between vertices  and . A radio labeling of a graph  is labeling using distinct nonnegative integers such that  for every pair of distinct vertices ,  in the vertex set of . Then the radio number of , commonly denoted , is the smallest integer  such that  has a radio labeling  with .

The smallest integer  such that  has a radio labeling  with  is known as the radio number of , commonly denoted .

Define . A radio labeling  of  is optimal if .