• Chung, Soon-Yeong
  • Received : 2009.02.20
  • Published : 2010.11.01


The purpose of this work is to identify the internal structure of the electrical networks with data obtained from only a part of network or the boundary of network. To be precise, it is discussed whether we can identify resistors or electrical conductivities of each link inside networks by the measurement of voltage on the boundary which is induced by a prescribed current on the boundary. As a result, it is shown that the structure of the resistor network can be determined uniquely by only one pair of the data (current, voltage) on the boundary, if the resistors satisfy an appropriate condition. Besides, several useful results about the energy functionals, which means the electrical power, are included.


resistor network;Thompson's principle;inverse problem


  1. G. Alessandrini, Remark on a paper by H. Bellout and A. Friedman, Boll. Un. Mat. Ital. A (7) 3 (1989), no. 2, 243-249.
  2. H. Bellout and A. Friedman, Identification problems in potential theory, Arch. Rational Mech. Anal. 101 (1988), no. 2, 143-160.
  3. N. L. Biggs, Algebraic Graph Theory, Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1993.
  4. N. L. Biggs, Potential theory on distance-regular graphs, Combin. Probab. Comput. 2 (1993), no. 3, 243-255.
  5. N. L. Biggs, Algebraic potential theory on graphs, Bull. London Math. Soc. 29 (1997), no. 6, 641-682.
  6. J. A. Bondy and R. L. Hemminger, Graph reconstruction|a survey, J. Graph Theory 1 (1977), no. 3, 227-268.
  7. A. P. Calderon, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pp. 65-73, Soc. Brasil. Mat., Rio de Janeiro, 1980.
  8. F. R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathemat-ics, 92. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997.
  9. F. R. K. Chung, M. Garrett, R. Graham, and D. Shallcross, Distance realization problems with applications to internet tomography, J. Comput. System Sci. 63 (2001), 432-448.
  10. F. R. K. Chung and R. P. Langlands, A combinatorial Laplacian with vertex weights, J. Combin. Theory Ser. A 75 (1996), no. 2, 316-327.
  11. F. R. K. Chung and K. Oden, Weighted graph Laplacians and isoperimetric inequalities, Pacific J. Math. 192 (2000), no. 2, 257-273.
  12. F. R. K. Chung and S.-T. Yau, Discrete Green's functions, J. Combin. Theory Ser. A 91 (2000), no. 1-2, 191-214.
  13. D. M. Cvetkovic, M. Doob, I. Gutman, and A. Torgasev, Recent Results in The Theory of Graph Spectra, Annals of Discrete Mathematics, 36. North-Holland Publishing Co., Amsterdam, 1988.
  14. P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, Carus Mathematical Monographs, 22. Mathematical Association of America, Washington, DC, 1984.
  15. D. M. Cvetkovic, M. Doob, and H. Sachs, Spectra of Graphs, Theory and application. Pure and Applied Mathematics, 87. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.
  16. S. L. Hakimi and S. S. Yau, Distance matrix of a graph and its realizability, Quart. Appl. Math. 22 (1965), 305-317.
  17. V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127. Springer-Verlag, New York, 1998.
  18. V. Isakov and J. Powell, On the inverse conductivity problem with one measurement, Inverse Problems 6 (1990), no. 2, 311-318.
  19. J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153-169.

Cited by

  1. On the Solvability of the Discrete Conductivity and Schrödinger Inverse Problems vol.76, pp.3, 2016,
  2. Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams pp.1572-9036, 2018,