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 one-way infinite paths. As a result, a variety of difficulties arise in that theory, leading to the need to restrict such path-based graphs in various ways to obtain certain results. In this work, we present a more general theory of transfinite graphs, wherein connectedness and the designation of extremities are accomplished through walks rather than paths. This leads to a simpler and yet more general theory, wherein new kinds of transfinite extremities are also encompassed. For instance, an ordinal-valued distance function can now be defined on all pairs of walk-connected nodes, in contrast to the path-based theory, wherein no distance function is definable for those pairs of nodes that are not path connected even though they are walk connected. Some results concerning eccentricities, radii, and diameters are presented in this more general walk-based graph theory. Another new result herein is the development of an electrical network theory for networks whose graphs are walk based. A unique voltage-current regime is established under certain conditions. The current regime is built up from current flows in closed transfinite walks—in contrast to a prior theory based upon flows in transfinite loops. A notable advantage of the present approach is that node voltages with respect to a given ground node are always unique whenever they exist. The present approach is more general in that it provides nontrivial voltage-current regimes for certain networks for which the prior approach would only provide trivial solutions having only zero currents and voltages everywhere.