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In a square product Kn x Kn there are two channel graphs, pertaining respectively to vertex pairs which are within a subgraph isomorphic to Kn and to those which are not. We take as an example the latter (which is the majority case if n > 3). It follows from properties (iii) and (iv) above that the channel graph comprises all vertices and edges of the product graph (it differs only in that two vertices are labeled as terminals), unless some further restriction be made. In practice, one would wish to simplify control and economize in the use of links; both these ends seem likely to be achieved by restricting the path length.

CATTERMOLE (d) Open parallel :/,) Fig. 17 A Takagi graph may be defined as a t-stage graph derived by parallel­ ing operations from the unit graph = C( 1, 1, . . , 1). The importance of this construction resides in the following proper­ ties: the operations are commutative, they yield graphs which are 1 2 3 • ------------ • ---------------• = (unit graph) G 2 = P (1,3 :m2:Gi) m2 (*,2 :m ,:G2) m, m2 C4= P (2,*:m3:G3) 3 Fig. 18 2 GRAPH THEORY AND COMMUNICATIONS NETWORKS 33 channel-regular, and the class of graphs whose properties can be estab­ lished by the method includes many of practical interest and utility.