Transfinite graphs and electrical networks

Author:
A. H. Zemanian

Journal:
Trans. Amer. Math. Soc. **334** (1992), 1-36

MSC:
Primary 94C15; Secondary 05C90

DOI:
https://doi.org/10.1090/S0002-9947-1992-1066452-1

MathSciNet review:
1066452

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: All prior theories of infinite electrical networks assume that such networks are finitely connected, that is, between any two nodes of the network there is a finite path. This work establishes a theory for transfinite electrical networks wherein some nodes are not connected by finite paths but are connected by transfinite paths. Moreover, the voltages at those nodes may influence each other. The main difficulty to surmount for this extension is the construction of an appropriate generalization of the concept of connectedness. This is accomplished by extending the idea of a node to encompass infinite extremities of a graph. The construction appears to be novel and leads to a hierarchy of transfinite graphs indexed by the finite and infinite ordinals. Two equivalent existence and uniqueness theorems are established for transfinite resistive electrical networks based upon Tellegen's equation, one using currents and the other using voltages as the fundamental quantities. Kirchhoff's laws do not suffice for this purpose and indeed need not hold everywhere in infinite networks. Although transfinite countable electrical networks have in general an uncountable infinity of extremities, called "tips," the number of different tip voltages may be radically constrained by both the graph of the network and its resistance values. Conditions are established herein under which various tip voltages are compelled to be the same. Furthermore, a theorem of Shannon-Hagelbarger on the concavity of resistance functions is extended to the driving-point resistance between any two extremities of arbitrary ranks. This is based upon an extension of Thomson's least power principle to transfinite networks.

**[1]**L. De Michèle and P. M. Soardi,*A Thomson's principle for infinite, nonlinear, resistive networks*, Proc. Amer. Math. Soc.**109**(1992), 461-468.**[2]**V. Dolezal,*Nonlinear networks*, Elsevier, New York, 1977. MR**0490486 (58:9831)****[3]**-,*Monotone operators and applications in control and network theory*, Elsevier, New York, 1979.**[4]**V. Dolezal and A. H. Zemanian,*Hilbert networks*II:*Some qualitative properties*, SIAM J. Control**13**(1975), 153-161. MR**0363689 (50:16126b)****[5]**P. G. Doyle,*Electric currents in infinite networks*, preprint, 1988.**[6]**P. G. Doyle and J. L. Snell,*Random walks and electric networks*, The Cams Mathematical Monographs, Math. Assoc. Amer., Washington, D.C., 1984. MR**920811 (89a:94023)****[7]**H. Flanders,*Infinite networks*: -*Resistance networks*, IEEE Trans. Circuit Theory**CT-18**(1971), 326-331. MR**0275998 (43:1751)****[8]**R. Halin,*Some path problems in graph theory*, Abh. Math. Sem. Univ. Hamburg**44**(1975), 175-186. MR**0429640 (55:2651)****[9]**J. Jeans,*Electricity and magnetism*, 5th ed., Cambridge Univ. Press, London, 1927. MR**0115577 (22:6378)****[10]**L. A. Liusternik and V. J. Sobolev,*Elements of functional analysis*, Ungar, New York, 1961. MR**0141967 (25:5362)****[11]**H. M. Melvin,*On the concavity of resistance functions*, J. Appl. Phys.**27**(1956), 658-659.**[12]**C. St. J. A. Nash-Williams,*Random walks and electric currents in networks*, Proc. Cambridge Philos. Soc.**55**(1959), 181-194. MR**0124932 (23:A2239)****[13]**J. E. Rubin,*Set theory*, Holden-Day, San Francisco, Calif., 1967. MR**0215726 (35:6561)****[14]**E. Schlesinger,*Infinite networks and Markov chains*, preprint, 1989. MR**1164936 (93m:94051)****[15]**C. E. Shannon and D. W. Hagelbarger,*Concavity of resistance functions*, J. Appl. Phys.**27**(1956), 42-43.**[16]**F. Spitzer,*Principles of random walk*, Van Nostrand, Princeton, N.J., 1964. MR**0171290 (30:1521)****[17]**P. M. Soardi and W. Woess,*Uniqueness of currents in infinite resistive networks*, Discrete Appl. Math.**31**(1991), 37-49. MR**1097526 (92b:94052)****[18]**C. Thomassen,*Resistances and currents in infinite electrical networks*, J. Combin. Theory Ser. B**49**(1990), 87-102. MR**1056821 (91d:94029)****[19]**A. H. Zemanian,*Counlably infinite networks that need not be locally finite*, IEEE Trans. Circuits and Systems**CAS-21**(1974), 274-277. MR**0459983 (56:18171)****[20]**-,*Infinite networks of positive operators*, Circuit Theory and Applications**2**(1974), 69-78.**[21]**-,*Connections at infinity of a countable resistive network*, Circuit Theory and Applications**3**(1975), 333-337.**[22]**-,*Infinite electrical networks*, Proc. IEEE**64**(1976), 6-17. MR**0453371 (56:11635)****[23]**-,*The limb analysis of countably infinite electrical networks*, J. Combin. Theory Ser. B**24**(1978), 76-93. MR**0465596 (57:5494)****[24]**-,*Infinite electrical networks with finite sources at infinity*, IEEE Trans. Circuits and Systems**CAS-34**(1987), 1518-1534. MR**923475 (88m:94039)****[25]**-,*Infinite electrical networks*:*A reprise*, IEEE Trans. Circuits and Systems**CAS-35**(1988), 1346-1358. MR**964586 (89i:94054)****[26]**-,*Transfinite cascades*, IEEE Trans. Circuits and Systems**38**(1991), 78-85.**[27]**-,*Boundary conditions at infinity for a discrete form of*, , State University of New York at Stony Brook, CEAS Technical Report 523, August, 1988.**[28]**-,*Transfinite random walks based on electrical networks*, State University of New York at Stony Brook, CEAS Tech. Rep. 604, June 4, 1991.**[29]**-,*Random walks on*-*networks*, Harmonic Analysis and Discrete Potential Theory (M. Picardello, Editor), Plenum, London, 1992 (in press).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
94C15,
05C90

Retrieve articles in all journals with MSC: 94C15, 05C90

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1066452-1

Keywords:
Infinite electrical networks,
transfinite graphs,
transfinite connectedness,
currents at and beyond infinity

Article copyright:
© Copyright 1992
American Mathematical Society