Graphs and Networks

This book examines results on transfinite graphs and networks achieved through research over 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 preliminary theory summarizing all essential ideas needed, relieving the reader of any need to consult those prior books. Subsequent chapters are devoted entirely to novel results and cover: - Connectedness ideas and their relationship to hypergraphs - Distance ideas 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