Connectedness in transfinite graphs and the existence and uniqueness of node voltages Original Research Article

Unlike connectedness in ordinary graphs, transfinite connectedness need not be transitive. As a result, sections of a transfinite graph that are maximal with respect to transfinite connectedness may overlap while being different, as is shown by an example. A sufficient condition is established under which transitivity holds, in which case the said sections partition the transfinite graph. A related phenomenon is that it may not be possible to assign a unique voltage to a node of a transfinite electrical network because the sum of the branch voltages along a path between that node and a chosen ground node may depend upon the choice of the path. This too is shown by example. Sufficient conditions are established that insure that all nodes have unique node voltages, being independent of the choices of the paths to ground. The proofs are based on a characterization of the totally ordered set of nodes along any transfinite path, the characterization being a certain hierarchical structure of nested sequences.