Transfinite graphs and electrical networks

Author:
A. H. Zemanian

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

MSC:
Primary 94C15; Secondary 05C90

MathSciNet review:
1066452

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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]**Vaclav Dolezal,*Nonlinear networks*, Elsevier Scientific Publishing Co., Amsterdam-New York-Oxford, 1977. MR**0490486****[3]**-,*Monotone operators and applications in control and network theory*, Elsevier, New York, 1979.**[4]**Václav Doležal and Armen Zemanian,*Hilbert networks. II. Some qualitative properties*, SIAM J. Control**13**(1975), 153–161. MR**0363689****[5]**P. G. Doyle,*Electric currents in infinite networks*, preprint, 1988.**[6]**Peter G. Doyle and J. Laurie Snell,*Random walks and electric networks*, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984. MR**920811****[7]**Harley Flanders,*Infinite networks. I: Resistive networks*, IEEE Trans. Circuit Theory**CT-18**(1971), 326–331. MR**0275998****[8]**R. Halin,*Some path problems in graph theory*, Abh. Math. Sem. Univ. Hamburg**44**(1975), 175–186 (1976). MR**0429640****[9]**James Jeans,*The mathematical theory of electricity and magnetism*, 5th ed, Cambridge University Press, New York, 1960. MR**0115577****[10]**L. A. Liusternik and V. J. Sobolev,*Elements of functional analysis*, Russian Monographs and Texts on Advanced Mathematics and Physics, Vol. 5, Hindustan Publishing Corp., Delhi; Gordon and Breach Publishers, Inc., New York, 1961. MR**0141967****[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 walk and electric currents in networks*, Proc. Cambridge Philos. Soc.**55**(1959), 181–194. MR**0124932****[13]**Jean E. Rubin,*Set theory for the mathematician*, Holden-Day, Inc., San Francisco, Calif.-Cambridge-Amsterdam, 1967. MR**0215726****[14]**E. Schlesinger,*Infinite networks and Markov chains*, Boll. Un. Mat. Ital. B (7)**6**(1992), no. 1, 23–37 (English, with Italian summary). MR**1164936****[15]**C. E. Shannon and D. W. Hagelbarger,*Concavity of resistance functions*, J. Appl. Phys.**27**(1956), 42-43.**[16]**Frank Spitzer,*Principles of random walk*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1964. MR**0171290****[17]**Paolo M. Soardi and Wolfgang Woess,*Uniqueness of currents in infinite resistive networks*, Discrete Appl. Math.**31**(1991), no. 1, 37–49. MR**1097526**, 10.1016/0166-218X(91)90031-Q**[18]**Carsten Thomassen,*Resistances and currents in infinite electrical networks*, J. Combin. Theory Ser. B**49**(1990), no. 1, 87–102. MR**1056821**, 10.1016/0095-8956(90)90065-8**[19]**A. H. Zemanian,*Countably infinite networks that need not be locally finite*, IEEE Trans. Circuits and Systems**CAS-21**(1974), 274–277. MR**0459983****[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]**Armen H. Zemanian,*Infinite electrical networks*, Proc. IEEE**64**(1976), no. 1, 6–17. Recent trends in system theory. MR**0453371****[23]**A. H. Zemanian,*The limb analysis of countably infinite electrical networks*, J. Combinatorial Theory Ser. B**24**(1978), no. 1, 76–93. MR**0465596****[24]**A. H. Zemanian,*Infinite electrical networks with finite sources at infinity*, IEEE Trans. Circuits and Systems**34**(1987), no. 12, 1518–1534. MR**923475**, 10.1109/TCS.1987.1086090**[25]**A. H. Zemanian,*Infinite electrical networks: a reprise*, IEEE Trans. Circuits and Systems**35**(1988), no. 11, 1346–1358. MR**964586**, 10.1109/31.14459**[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).

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Additional Information

DOI:
http://dx.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