Graphs and Networks (PDF)

This self-contained book examines results on transfinite graphs and networks achieved through a continuing research effort during the past several years. These new results, covering the mathematical theory of electrical circuits, are different from those presented in two previously published books by the author, Transfiniteness for Graphs, Electrical Networks, and Random Walks and Pristine Transfinite Graphs and Permissive Electrical Networks.

Two initial chapters present the preliminary theory summarizing all essential ideas needed for the book and will relieve the reader from any need to consult those prior books. Subsequent chapters are devoted entirely to novel results and cover:

* Connectedness ideas---considerably more complicated for transfinite graphs as compared to those of finite or conventionally infinite graphs----and their relationship to hypergraphs

* Distance ideas---which play an important role in the theory of finite graphs---and their extension to transfinite graphs with more complications, such as the replacement of natural-number distances by ordinal-number distances

* Nontransitivity of path-based connectedness alleviated by replacing paths with walks, leading to a more powerful theory for transfinite graphs and networks

Additional features include:

* The use of nonstandard analysis in novel ways that leads to several entirely new results concerning hyperreal operating points for transfinite networks and hyperreal transients on transfinite transmission lines; this use of hyperreals encompasses for the first time transfinite networks and transmission lines containing inductances and capacitances, in addition to resistances

* A useful appendix with concepts from nonstandard analysis used in the book

* May serve as a reference text or as a graduate-level textbook in courses or seminars

Graphs and Networks: Transfinite and Nonstandard will appeal to a diverse readership, including graduate students, electrical engineers, mathematicians, and physicists working on infinite electrical networks. Moreover, the growing and presently substantial number of mathematicians working in nonstandard analysis may well be attracted by the novel application of the analysis employed in the work.