AbstractPart of this work is concerned with determining those graphs that produce extreme rays of the cone of shortest distance matrices of undirected linear n‐vertex graphs without loops. For n ≤ 4, for example, all extreme rays come from a certain class of “elementary” graphs, but for n ≥ 5 there are nonelementary extreme rays, and these are studied further. Another part of this work is concerned with the representation of a distance matrix as a set of distances among points in a normed vector space. The set of graphs (distance matrices) representable by a given norm and the set of norms (and dimensions of spaces) by which a given graph may be represented are studied. It is shown that lx is unique among lp norms in that any graph is lx representable in Rn−1. The l1 representable distance matrices are found to be just those in the cone generated by elementaries for all n. Other results on distance matrices are also included.