A concept of metric is defined in the set of labelled graphs. The concept is based on the idea that the extent of the difference in two graphs may be measured by the non-orthogonality of tie-set (loop) vectors of one graph to the cut-set vectors of the other. The meanings involved in the concept are discussed on the topics concerning flow and tension networks. Applications to electrical networks include the theorem that, for a non-reciprocal n-port network N which is realized by using λN controlled-sources, an inequality 1/2 rank (YYT)D(gc, gv)λN holds where Y is the admittance matrix of N and D(gc, gv) is the distance between the corresponding current graph gc and voltage graph gv of N. An example for which the equalities hold in the above inequality is given.