Abstract.The results of Hakimi and Yau and others in the realization of a distance matrix are generalized to graphs (digraphs) whose branches (arcs) may have negative weights. Conditions under which such matrices have a tree, hypertree or directed tree realization are given, uniqueness of these realizations is discussed and algorithms for their construction are indicated.1. Notation. A number of definitions are given so that results will be presented in The degree of a vertex Vi in G, denoted deg (y, , G), is the number of branches (arcs) incident at i>, in G. The outdegree of a vertex v{ in digraph G, denoted outdeg(y, , G), is equal to the number of arcs incident at v, in G and directed away from v{. The indegree of Vi, denoted indeg(z\-, G), is equal to the number of arcs incident at v{ in G and directed towards v{ . A weighted graph (digraph) is a graph (digraph) together with a function which assigns a real number wx to each branch h, (arc a,). All graphs (digraphs) presented here are weighted.An edge-sequence in a graph (digraph) between two vertices v, and vf is an alternating sequence of vertices and branches (arcs)• • • M,-beginning and ending with Vi and Vj , in which each branch (arc) is incident at the vertex preceding and the vertex following it. A path from to v,-is the set of all branches (arcs) in an edge-sequence between v{ and v, . A directed path in a digraph is a path in which each arc is directed from the vertex preceding it to the vertex following it in the corresponding edge-sequence. A path or directed path is called elementary if all vertices in the edge-sequence are distinct. A path (directed path) is a circuit {cycle) if the first and last vertex in the edgesequence are the same and all others distinct. The length of a path (directed path) is the sum of the weights of the branches (arcs) in it. A connected graph (digraph) is a graph (digraph) in which every pair of vertices is joined by a path.

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