Graphs and Networks: Transfinite and Nonstandard

206 Pages·2004·5.218 MB·English

by  Armen H. Zemanian (auth.)

For
Steven Michael
Carolyn Kearns
Laura Ann
Ann Odell
Kate Armine
Cormac Micheal
Armen H. ZemanÎan
Graphs and Ne tworks
Transfinite and Nonstandard
Springer Science+Business Media, LLC
Armen H. Zemanian
Department of Electrica1 Engineering
State University of New York at Stony Brook
Stony Brook, NY 11794-2350
U.S.A.
Library of Congress Cataloging-in-Publication Data
Zemanian, A. H. (Armen H.)
Graphs and networks : transfinite and nonstandard / Armen H. Zemanian.
p.em.
Includes bibliographieaI referenees and indexes.
ISBN 978-0-8176-4292-1 ISBN 978-0-8176-8178-4 (eBook)
DOI 10.1007/978-0-8176-8178-4
1. Graphy theory. 2. Transfinite numbers. 1. Title.
QAI66.z4472004
511' .5-de22 2004043745
CIP
AMS Subjeet Classifications: 94C05, 03H05. 05C40, 05C12, 03HIO, 04AIO
ISBN 978-0-8176-4292-1 Printed on acid-free paper.
© Springer Seienee+ Business Media New York 2004
Originally published by Birkhăuser Boston in 2004
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Contents
Preface.... .... ... . ... ... ... .. . ... .... ... .. ...... .... ... .... .... ix
1 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Concerning Symbols and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Ranks of Transfiniteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Transfinite Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Branches or Synonymously (-1 )-Graphs ........ . . . . . . . . . . . . . . . . 5
2.2 O-Graphs................................................... 6
2.3 I-Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 t.t-Graphs................................................... 10
2.5 iV-Graphs.......................... . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 w-Graphs................................................... 17
2.7 A Concise Characterization of Transfinite Paths and Loops . . . . . . . . . 19
2.8 Graphs of Higher Ranks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.9 Why Not Restrict "Extremities" to "Ends"? . . . . . . . . . . . . . . . . . . . . .. 22
3 Connectedness, Trees, and Hypergraphs .......................... 23
3.1 Transfinite Connectedness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
3.2 Transfinite Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Hypergraphs from v-Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Ordinal Distances in Transfinite Graphs .......................... 39
4.1 Natural Sums of Ordinals ..................................... 40
4.2 Lengths of Paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40
4.3 Metrizable Sets of Nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42
4.4 Distances between Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44
4.5 Eccentricities and Related Ideas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45
4.6 Some General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48
4.7 When the Nodes of Highest Rank Are Pristine.... ............ .... 52
4.8 The Center Lies in a v-Block .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57
4.9 The Centers of Cycle-free v-Graphs ............................ 60
Vi Contents
5 Walk-Based Transfinite Graphs and Networks . . . . . . . . . . . . . . . . . . . .. 63
5.1 O-Walks and 1-Wgraphs ...................................... 64
5.2 I-Walks, 2-Wgraphs, and 2-Walks.............................. 66
5.3 JL-Walks and (JL + 1)-Wgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 w-Wgraphs and w-Walks.......... ... .......... ............... 76
5.5 w-Wgraphs and w-Walks...................................... 79
5.6 Walk-based Extremities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.7 Lengths of Walks ............................................ 84
5.8 Wdistances between Wnodes ... ............................... 85
5.9 Weccentricities and Related Ideas .............................. 86
5.10 Walk-based Transfinite Electrical Networks . . . . . . . . . . . . . . . . . . . . . . 89
5.11 Tours and Tour Currents ...................................... 90
5.12 The Solution Space T ......... ...... .................... ..... 90
5.13 The Existence of a Unique Current-Voltage Regime............... 91
5.14 Kirchhoff's Laws.......................................... .. 93
5.15 The Uniqueness ofWnode Voltages............................. 95
6 Hyperreal Currents and Voltages in Transfinite Networks. . . . . . . . . . .. 99
6.1 Two Examples .............................................. 100
6.2 Restorable Networks ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 105
6.3 Hyperreal Currents and Voltages; A Hyperreal Operating Point. . . . .. 112
6.4 Eventual Connectedness, Eventual Separability, and Kirchhoff's Laws 113
6.5 Three Examples Involving Ladder Networks ..................... 118
6.6 Random Walks on Restorable Transfinite Networks ............... 122
6.7 Appending and Inserting Branches; Buildable Graphs .............. 128
6.8 Other Ideas: Nonstandard Graphs and Networks .................. 130
7 Hyperreal Transients in Transfinite RLC Networks . . . . . . . . . . . . . . . .. 133
7.1 Hyperreal Transients on the Hyperreal Time Line . . . . . . . . . . . . . . . .. 134
7.2 Hyperreal Transients in Restorable RLC Networks ................ 135
7.3 A Transfinite RLC Ladder ..................................... 135
7.4 A Transfinite Artificial Cable . .. . .. . . . . . . . . .. . . . .. . .. . .. . . . .. .. 138
7.5 A Transfinite Artificial Transmission Line ....................... 141
7.6 Conventionally Infinite, Uniform Transmission Lines and Cables
and Nonstandard Enlargements ................................ 143
7.7 The w2-Line ................................................ 145
7.8 A Hyperreal Wave on an w2-Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 146
7.9 Transfinite Lines of Higher Ranks .. .. . . . . . . . . . . . . . . .. .. . . . . .. .. 148
7.10 A Hyperreal Diffusion on a Transfinite Cable ..................... 151
8 Nonstandard Graphs and Networks .............................. 153
8.1 Nonstandard Graphs Defined .................................. 154
8.2 Incidences and Adjacencies between Nodes and Branches. . . . . . . . .. 156
8.3 Nonstandard Hyperfinite Paths and Loops ....................... 157
8.4 Connected Nonstandard Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 158
Contents Vll
8.5 Nonstandard Subgraphs ....................................... 159
8.6 Nonstandard Trees ........................................... 160
8.7 Some Numerical Formulas .................................... 160
8.8 Nonstandard I-Graphs ........................................ 163
8.9 A Fundamental Theorem for Nonstandard I-Networks ............. 164
A Some Elements of Nonstandard Analysis .......................... 171
B The Fibonacci Numbers ........................................ 187
C A Laplace Transform for an Artificial RC Cable .................... 189
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 191
Index of Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 195
Subject Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 197
Preface
Scientia Gratiii Scientiae
It is now thirteen years since the first book that discusses transfinite graphs and elec
trical networks appeared [50]. This was followed by two more books [51] and [54]
which compiled results from an ongoing research effort on that subject. Why then is
a fourth book, this one, being offered? Simply because still more has been achieved
beyond that appearing in those prior books. An exposition of these more recent re
sults is the purpose of this book.
The idea of transfiniteness for graphs and networks appeared as virgin research
territory about seventeen years ago. Notwithstanding the progress that has since been
achieved, much more remains to be done-or so it appears. Many conclusions con
cerning conventionally infinite graphs and networks can be reformulated as open
problems for transfinite graphs and networks. Furthermore, questions peculiar to
transfinite concepts for graphs and networks can be suggested. Indeed, these two
considerations have inspired the new results displayed herein.
There are presently four nonequivalent ways of introducing transfiniteness for
graphs and networks. The original method is discussed in [50], with [48] being the
earliest reconnaissance into that approach. It views the extremities of conventionally
infinite graphs as equivalence classes, called "tips," of one-way infinite paths that
are pairwise viewed as "equivalent" if they are pairwise eventually identical; con
nections between those tips are implemented by "I-nodes," these being transfinite
generalizations of conventional nodes. Such ideas can be extended to higher ranks
of transfiniteness and are examined at length in [51]. A concise exposition of this
approach is presented in Chapter 2.
It turns out that transfinite connectedness is not in general a transitive binary
relation between nodes. Specifically, one node may be connected to another node
through a transfinite path, and the second node may be connected to a third node
through another transfinite path, but there may be no transfinite path connecting the
first and third nodes. This pathology can be avoided by imposing some additional re
strictions on the transfinite graph. On the other hand, this difficulty disappears if the
extremities (now called "walk-tips" or simply "wtips") of a graph are represented by
equivalence classes of one-way infinite walks, rather than paths, with the walk-based
x Preface
tips being used to construct walk-based transfinite nodes. This radically expands the
realm of transfinite graphs but also introduces more complicated-indeed, strange
structures as walk-based transfinite graphs. The restrictions necessitated by the non
transitivity of path-connectedness are no longer needed for walk-connectedness.
Chapter 5 herein presents this walk-based approach to transfinite graphs.
The path-based and walk-based transfinite graphs are strictly graph-theoretic
constructs. Nonetheless, by assigning electrical parameters to the branches, trans
finite networks and their electrical behavior can be explored.
For instance, a different approach, due to B.D. Calvert, becomes feasible when
the branches are assigned positive resistances and yields another theory for transfinite
electrical networks. The sums of branch resistances in paths provide path lengths,
which lead to a metric for the set of nodes of a conventionally infinite, locally finite
graph. That metric space is discrete. However, its completion may have limit points,
and these can be interpreted as the infinite extremities of the network. Connections
between those limit points then yield a transfinite network. This leads to a substantial
theory for transfinite electrical networks. Moreover, it can be extended to higher
ranks of transfiniteness wherein each rank has its own metric, lower-ranked metrics
being stronger than higher-ranked metrics. This version of transfinite networks is
examined at some length in the book [54], but it is not discussed herein. We mention
it now since it is one of the four extant methods for constructing transfinite networks.
The fourth and radically different approach to transfinite graphs is provided by
nonstandard analysis. The nodes of a conventionally infinite graph along with the
real numbers are taken as the individuals for a superstructure of sets, from which
ultrapower constructions yield "nonstandard nodes," "nonstandard branches," and
thereby "nonstandard graphs." This is explored in Chapter 8, wherein "nonstandard
transfinite graphs and networks" are also proposed. Standard theorems concerning
the existence and uniqueness of current-voltage regimes in transfinite electrical net
works can be lifted into a nonstandard setting with now hyperreal currents and volt
ages.
Actually, byperreal currents and voltages can be introduced into transfinite net
works without enlarging those networks in a nonstandard way; this is the subject of
Chapter 6. The idea is to view the transfinite network as the end result of an expand
ing sequence of finite networks that fill out the transfinite network in such a way that
the connections provided by the transfinite nodes in the original network are main
tained throughout the expansion procedure. This yields a sequence of currents in each
branch, which can be identified as a hyperreal current for that branch, and similarly
for the branch and node voltages. Thus, the transfinite network obtains a hyperreal
current-voltage regime, which satisfies Kirchhoff's laws and Ohm's law, again in a
particular nonstandard way. Here, however, the transfinite network will have many
different hyperreal current-voltage regimes due to the many different ways of filling
out the network with sequences of finite networks. Another advantage is that trans
finite networks having inductors and capacitors in addition to resistors can now be
admitted. (All the prior approaches to transfinite networks are restricted to purely
resistive networks.) Thus, we can now examine hyperreal-valued transients in hy-
Preface xi
perreal time, as is done in Chapter 7. As a particular result of this sort, hyperreal
transients on artificial and distributed, transfinite, transmission lines and cables are
established.
Let us also mention that a fifth method of defining transfinite graphs can be de
vised by starting with the "ends" of conventionally infinite graphs, as defined by
Halin [25] and others. However, the restriction of the infinite extremities of conven
tional graphs to their ends turns out to be too crude for many purposes. For example,
a one-way infinite ladder has only one end, but connections to different extremities
represented by the tips within that single end are needed for electrical analysis. This
issue is discussed in Sec. 2.9.
As for the other chapters of this book, some symbols and notations are specified
and transfinite ranks are explicated in Chapter 1. Chapter 2 is a concise presenta
tion of the earliest construction of transfinite graphs, a topic that is explored at some
length in [51]. Chapter 2 provides enough information to make this book understand
able independently of that prior book. In Chapter 3, some issues regarding transfinite
connectedness are examined, and transfinite graphs are related to hypergraphs in a
certain way.
Furthermore, distances between nodes have played an important role in graph
theory. Those distances are usually natural numbers. Chapter 4 extends that subject
to transfinite graphs by allowing those distances to be transfinite ordinals as well. A
variety of standard results are extended transfinitely in this way.
This then is the scope of our book. But perhaps a few words about some general
properties and peculiarities of transfinite graphs and networks might be worthwhile.
Their definitions can be quite complicated. For example, a path in a conventional
graph can be defined in a sentence or two, whereas the definition of a transfinite path
of natural-number rank requires half a page along with another page or so of some
preliminary specifications of terms used in that definition. Moreover, since paths
are defined recursively along increasing transfinite ranks, with the arrow-rank paths
having still another distinctive definition, the whole structure is rather cumbersome.
To be sure, a concise characterization of transfinite paths is given in Sec. 2.7, but
it requires that the lengthy definitions of transfinite paths be established first. This
complexity is typical in all of the four different kinds of transfinite graphs and net
works. Would that there were a more concise definition of transfinite paths. Perhaps
there is, but it is yet to be devised.
One last comment: Transfinite numbers were introduced by Cantor into mathe
matics over 100 years ago, and this led to a major development in the set-theoretic
foundations of mathematics. This has expanded the scope of a variety of subjects in
modem mathematics. In contrast, graph theory has remained on "this side of infin
ity," notwithstanding the substantial body of results concerning conventionally in
finite graphs. Undoubtedly, the advent of transfinite graphs will not by any means
have the same impact as have transfinite numbers. The subject is for the most part
isolated from other areas of mathematics-in contrast to conventional graph theory,
but perhaps this may change. It would change if practical applications to engineer
ing and science were to be found, but nothing of this nature exists presently. All