UNCOUNTABLE TRANSFINITE GRAPHS AND ELECTRICAL NETWORKS

Up to now the theory of infinite and transfinite graphs and electrical networks has been established only for the case where the branch sets are countable. This work extends that prior theory to graphs and netw orks with uncountable branch sets, a structure that arises quite natur ally in the theory of transfinite graphs. Moreover, conditions on the local structure of the transfinite graph are given which insure the co untability of the branch set. Finally, it is shown that only countably many of the branches in an uncountable electrical network can have no n-zero voltages and currents, and therefore the restriction to countab ility in the prior theory is not essentially restrictive.