Graphs and Networks

• 1 Some Preliminaries.- 1.1 Concerning Symbols and Terminology.- 1.2 Ranks of Transfiniteness.- 2 Transfinite Graphs.- 2.1 Branches or Synonymously (-l)-Graphs.- 2.2 0-Graphs.- 2.3 1-Graphs.- 2.4 ?-Graphs.- 2.5% MathType!MTEF!2!1!+-% feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGafqyYdCNbaSaaaaa!3CAA!$$\vec \omega$$-Graphs.- 2.6 ?-Graphs.- 2.7 A Concise Characterization of Transfinite Paths and Loops.- 2.8 Graphs of Higher Ranks.- 2.9 Why Not Restrict “Extremities” to “Ends”?.- 3 Connectedness, Trees, and Hypergraphs.- 3.1 Transfinite Connectedness.- 3.2 Transfinite Trees.- 3.3 Hypergraphs from ?-Graphs.- 4 Ordinal Distances in Transfinite Graphs.- 4.1 Natural Sums of Ordinals.- 4.2 Lengths of Paths.- 4.3 Metrizable Sets of Nodes.- 4.4 Distances between Nodes.- 4.5 Eccentricities and Related Ideas.- 4.6 Some General Results.- 4.7 When the Nodes of Highest Rank Are Pristine.- 4.8 The Center Lies in a ?-Block.- 4.9 The Centers of Cycle-free ?-Graphs.- 5 Walk-Based Transfinite Graphs and Networks.- 5.1 0-Walks and 1-Wgraphs.- 5.2 1-Walks, 2-Wgraphs, and 2-Walks.- 5.3 ?-Walks and (? + 1)-Wgraphs.- 5.4% MathType!MTEF!2!1!+-% feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGafqyYdCNbaSaaaaa!3CAA!$$\vec \omega$$-Wgraphs and% MathType!MTEF!2!1!+-%feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGafqyYdCNbaSaaaaa!3CAA!$$\vec \omega$$-Walks.- 5.5 ?-Wgraphs and ?- Walks.- 5.6 Walk-based Extremities.- 5.7 Lengths of Walks.- 5.8 Wdistances between Wnodes.- 5.9 Weccentricities and Related Ideas.- 5.10 Walk-based Transfinite Electrical Networks.- 5.11 Tours and Tour Currents.- 5.12 The Solution Space T.- 5.13 The Existence of a Unique Current-Voltage Regime.- 5.14 Kirchhoff’s Laws.- 5.15 The Uniqueness of Wnode Voltages.- 6 Hyperreal Currents and Voltages in Transfinite Networks.- 6.1 Two Examples.- 6.2 Restorable Networks.- 6.3 Hyperreal Currents and Voltages; A Hyperreal Operating Point.- 6.4 Eventual Connectedness, Eventual Separability, and Kirchhoff’s Laws.- 6.5 Three Examples Involving Ladder Networks.- 6.6 Random Walks on Restorable Transfinite Networks.- 6.7 Appending and Inserting Branches; Buildable Graphs.- 6.8 Other Ideas: Nonstandard Graphs and Networks.- 7 Hyperreal Transients in Transfinite RLC Networks.- 7.1 Hyperreal Transients on the Hyperreal Time Line.- 7.2 Hyperreal Transients in Restorable RLC Networks.- 7.3 A Transfinite RLC Ladder.- 7.4 A Transfinite Artificial Cable.- 7.5 A Transfinite Artificial Transmission Line.- 7.6 Conventionally Infinite, Uniform Transmission Lines and Cables and Nonstandard Enlargements.- 7.7 The ?2-Lmc.- 7.8 A Hyperreal Wave on an ?2-Line.- 7.9 Transfinite Lines of Higher Ranks.- 7.10 A Hyperreal Diffusion on a Transfinite Cable.- 8 Nonstandard Graphs and Networks.- 8.1 Nonstandard Graphs Defined.- 8.2 Incidencesand Adjacencies between Nodes and Branches.- 8.3 Nonstandard Hyperfinite Paths and Loops.- 8.4 Connected Nonstandard Graphs.- 8.5 Nonstandard Subgraphs.- 8.6 Nonstandard Trees.- 8.7 Some Numerical Formulas.- 8.8 Nonstandard 1-Graphs.- 8.9 A Fundamental Theorem for Nonstandard 1-Networks.- A SomeElements of Nonstandard Analysis.- B The Fibonacci Numbers.- C A Laplace Transform for an Artificial RC Cable.- References.- Index of Symbols.

"The book presents almost everything that has been achieved so far in the relatively new field of transfinite graphs and electrical networks. In the first two chapters, the reader is familiarized with transfinite graphs and with the symbols and notations used in the book... The last chapter is dedicated to the approach of nonstandard analysis applied to transfinite graphs. The book must be appreciated especially because new results in the field of transfinite graphs are also included. Therefore, I find this book a welcome addition to the literature." -Zentralblatt MATH "For about thirty years Zemanian has been developing a theory of infinite electrical networks. This book is the latest in a series of books...on the subject. The subject is necessarily abstract and sophisticated because infinite objects are the main objects of discourse... The first few chapters are important not only to remind the reader of the terms, but also to give an improved or alternate treatment of some earlier results... There does not yet seem to be a large following of researchers in this area, but it seems very attractive and ripe for investigation. It's intriguing to see the connections between set theory and electrical network problems... To understand these concepts fully the reader must consult the book under review. The reviewer highly recommends devoting the effort needed to understand these original and surprising concepts." -SIAM Review