A note on a matrix criterion for unique colorability of a signed graph, Psychometrika

A signed graph, S, is colorable if its point set can be partitioned into subsets such that all positive lines join points of the samsubset and all negative lines join points of different subsets. S is uniquely colorable if there is only one such partition. Developed in this note is a new matrix, called the type matrix of S, which provides a classification f the way pairs of points are joined in S. Such a classification yields a criterion f r color-ability and unique colorability. Much work has appeared in recent years exploring the properties of natural clusterings in structures of various sorts. One branch of this work has stemmed from the thinking of Heider [1946, 1958], who suggested certain rules that seem to govern the arrangement of positive and negative affective relationships among cognitive lements. Heider described certain structures as being "balanced " and postulated that these would be preferred over other structures which were said to be "imbalanced. " This formulation, stated in terms of "elements " and "signed relationships, " suggested to Cartwrigh