Ordinal Distances in Transfinite Graphs

An ordinal-valued metric, taking its values in the set א1 of all countable ordinals, can be assigned to a metrizable set M of nodes in any transfinite graph. M contains all the nonsingleton nodes, as well as certain singleton nodes. Moreover, this yields a graphical realization of Cantor’s countable ordinals, as well as of the Aristotelian ideas of “potential” and “actual” infinities, the former being represented by the arrow ranks and the latter by the ordinal ranks of transfiniteness. This construct also extends transfinitely the ideas of nodal eccentricities, radii, diameters, centers, peripheries, and blocks for graphs, and the following generalizations are established. With ν denoting the rank of a ν-graph G , the ν-nodes of G comprise the center of a larger ν-graph. Also, when there are only finitely many ν-nodes and when those ν-nodes are “pristine” in the sense that they do not embrace nodes of lower ranks, the infinitely many nodes of all ranks have eccentricities of the form ω · p, where ω is the first transfinite ordinal and p lies in a finite set of natural numbers. Furthermore, the center is contained in a single block of highest rank. Also, when each loop of the ν-graph is confined within a (ν − 1)-section, the center either is a single node of highest rank, or is the set of internal nodes of a (ν − 1)-section, or is the union of the latter two kinds of centers.