Uncountable transfinite graphs and electrical networks
A. H. Zemanian · International Journal of Circuit Theory and Applications · 1994
Abstract 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 networks with uncountable branch sets, a structure that arises quite naturally in the theory of transfinite graphs. Moreover, conditions on the local structure of the transfinite graph are given which insure the countability of the branch set. Finally, it is shown that only countably many of the branches in an uncountable electrical network can have non‐zero voltages and currents, and therefore the restriction to countability in the prior theory is not essentially restrictive.