Distance matrix of a graph and its realizability

The distances in a linear graph are described by a distance matrix D D . The realizability of a given D D by a linear graph is discussed and conditions under which the realization of D D is unique are established. The optimum realization of D D , (i.e., the realization of D D with “minimum total length"), is investigated. A procedure is given by which a tree realization of D D can be found, if such a realization exists. Finally, it is shown that a tree realization, if it exists, is unique and is the optimum realization of D D .