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Potential Theory on Distance-Regular Graphs

Published online by Cambridge University Press:  12 September 2008

Norman L. Biggs
Norman L. Biggs
Affiliation:
London School of Economics, Houghton St., London WC2A 2AE
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Abstract

A graph may be regarded as an electrical network in which each edge has unit resistance. We obtain explicit formulae for the effective resistance of the network when a current enters at one vertex and leaves at another in the distance-regular case. A well-known link with random walks motivates a conjecture about the maximum effective resistance. Arguments are given that point to the truth of the conjecture for all known distance-regular graphs.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 3 , September 1993 , pp. 243 - 255
Copyright
Copyright © Cambridge University Press 1993

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References

[1]Aldous, D. (1989) An introduction to covering problems for random walks on graphs. J. Theoretical Probability 2 87–120.CrossRefGoogle Scholar
[2]Biggs, N. L. (1974) Algebraic Graph Theory, Cambridge University Press. (Revised edition to be published in 1993.)CrossRefGoogle Scholar
[3]Bollobás, B. (1979) Graph Theory: An Introductory Course, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[4]Brouwer, A. E., Cohen, A. M. and Neumaier, A. (1989) Distance-Regular Graphs, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[5]Doyle, P. G. and Snell, J. L. (1984) Random Walks and Electrical Networks, Math. Assoc. of America.CrossRefGoogle Scholar
[6]Foster, R. M. (1949) The average impedance of an electrical network. In: Reissner Anniversary Volume – Contributions to Applied Mechanics, J. W. Edwards, Ann Arbor, Michigan333–340.Google Scholar
[7]Foster, R. M. (1961) An extension of a network theorem. IRE Trans. Circuit Theory CT-8 75–76.CrossRefGoogle Scholar
[8]Nerode, A. and Shank, H. (1961) An algebraic proof of Kirchhoff's network theorem. Amer. Math. Monthly 68 244–247.CrossRefGoogle Scholar
[9]Thomassen, C. (1990) Resistances and currents in infinite electrical networks. J. Comb. Theory B 49 87–102.CrossRefGoogle Scholar