Transfiniteness: For Graphs, Electrical Networks, and Random Walks

Transfinite numbers were invented by Cantor in the 19th century, and they profoundly affected the development of 20th-century mathematics. Despite this century-old introduction of transfiniteness into mathematics, its implications for graphs has only been examined recently. This text focuses on transfinite graphs and their ramifications for several related topics, such as electrical networks, discrete potential theory, and random walks. A transfinite graph is one wherein at least two nodes are connected through infinite paths but not through any finite path. Such a structure arises when two infinite graphs are joined at their infinite extremities by "1-nodes". In fact, infinitely many graphs may be so joined,and the result may have infinite extremities in a more general sense. The latter may in turn be joined by "2-nodes". These ideas can be extended recursively through the countable ordinals to obtain a hierarchy of transfinite graphs.