Methodology And Computing In Applied Probability

, Volume 6, Issue 4, pp 381–387 | Cite as

Foster's Formulas via Probability and the Kirchhoff Index

  • José Luis Palacios
Article

Abstract

Building on a probabilistic proof of Foster’s first formula given by Tetali (1994), we prove an elementary identity for the expected hitting times of an ergodic N-state Markov chain which yields as a corollary Foster’s second formula for electrical networks, namely
$$\sum {R_{ij} } \frac{{C_{iv} C_{vj} }}{{C_v }} = N - 2,$$
where Rij is the effective resistance, as computed by means of Ohm’s law, measured across the endpoints of two adjacent edges (i,v) and (i,v), Civ and Civ are the conductances of these edges, Cv is the sum of all conductances emanating from the common vertex v, and the sum on the left hand side of (1) is taken over all adjacent edges. We show how to extend Foster’s first and second formulas. As an application, we show how to use a “third formula” to compute the Kirchhoff index of a class of graphs with diameter 3.
effective resistance geodetic graphs 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. Bollobás, Modern Graph Theory, Springer: New York, 1998.Google Scholar
  2. A. K. Chandra, P. Raghavan, W. L. Ruzzo, R. Smolensky, and P. Tiwari, “The electrical resistance of a graph captures its commute and cover times,” In Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing, Seattle, Washington, pp. 574–586, 1989.Google Scholar
  3. A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Springer: Berlin, 1989.Google Scholar
  4. D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir, “Random walks on weighted graphs and applications to online algorithms,” In Proceedings of the 22nd Symposium on the Theory of Computing, pp. 369–378, 1990.Google Scholar
  5. P. G. Doyle and J. L. Snell, Random walks and electrical networks, The Mathematical Association of America: Washington, D.C., 1984.Google Scholar
  6. R. M. Foster, “The average impedance of an electrical network,” In Contributions to Applied Mechanics (Reissner Aniversary Volume); Edwards Brothers, Ann Arbor, MI, pp. 333–340, 1949.Google Scholar
  7. R. M. Foster, “An extension of a network theorem,” IRE Transactions on Circuit Theory vol. 8 pp. 75–76, 1961.Google Scholar
  8. D. Klein and M. Randić, “Resistance distance,” Journal of Mathematical Chemistry vol. 12 pp. 81–95, 1993.Google Scholar
  9. J. L. Palacios, “Resistance distance in graphs and random walks,” International Journal of Quantum Chemistry vol. 81 pp. 29–33, 2001a.Google Scholar
  10. J. L. Palacios, “Closed-form formulas for Kirchhoff index,” International Journal of Quantum Chemistry vol. 81 pp. 135–140, 2001b.Google Scholar
  11. P. Tetali, “Random walks and the effective resistance of networks,” Journal of Theoretical Probability vol. 4 pp. 101–109, 1991.Google Scholar
  12. P. Tetali, “An extension of Foster's network theorem,” Combinatories, Probability and Computing vol. 3 pp. 421–427, 1994.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • José Luis Palacios
    • 1
  1. 1.Departamento de Cómputo Científico y EstadísticaUniversidad Simón BolívarCaracasVenezuela

Personalised recommendations