Distance matrix of a graph and its realizability

Abstract

Abstract.The distances in a linear graph are described by a distance matrix D. The realizability of a given D by a linear graph is discussed and conditions under which the realization of D is unique are established. The optimum realization of D, (i.e., the realization of D with "minimum total length"), is investigated. A procedure is given by which a tree realization of 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.

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