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
.