Ah. Zemanian, UNCOUNTABLE TRANSFINITE GRAPHS AND ELECTRICAL NETWORKS, International journal of circuit theory and applications, 22(5), 1994, pp. 387-398
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.