Walk-Based Transfinite Graphs and Networks

Abstract

The theory of transfinite graphs developed so far has been based on the ideas that connectedness is accomplished through paths and that the infinite extremities of the graph are specified through equivalence classes of one-ended paths. This is a natural extension of finite graphs because connectedness for finite graphs is fully characterized by paths; indeed, any walk terminating at two nodes of a finite graph contains a path doing the same. However, such is no longer the case for transfinite graphs. Indeed, path-connectedness need not be transitive as a binary relationship among transfinite nodes, and Condition 3.1-2 was imposed to ensure such transitivity. Without that condition, distances as defined by paths do not exist between certain pairs of nonsingleton nodes. This limitation is also reflected in the theory of transfinite electrical networks by the fact that node voltages need not be uniquely determined when they are defined along paths to a chosen ground node.