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Resistance distance in wheels and fans

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Abstract

The wheel graph is the join of a single vertex and a cycle, while the fan graph is the join of a single vertex and a path. The resistance distance between any two vertices of a wheel and a fan is obtained. The resistances are related to Fibonacci numbers and generalized Fibonacci numbers. The derivation is based on evaluating determinants of submatrices of the Laplacian matrix. A combinatorial argument is also illustrated. A connection with the problem of squaring a rectangle is described.

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Author notes

  1. Somit Gupta

    Present address: David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

Authors and Affiliations

  1. Indian Statistical Institute, New Delhi, 110 016, India

    R. B. Bapat

  2. National Institute of Technology Karnataka, Surathkal, Mangalore, 575 025, India

    Somit Gupta

Authors
  1. R. B. Bapat
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  2. Somit Gupta
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Corresponding author

Correspondence to R. B. Bapat.

Additional information

The author gratefully acknowledges the support of the JC Bose Fellowship, Department of Science and Technology, Government of India.

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Bapat, R.B., Gupta, S. Resistance distance in wheels and fans. Indian J Pure Appl Math 41, 1–13 (2010). https://doi.org/10.1007/s13226-010-0004-2

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  • DOI: https://doi.org/10.1007/s13226-010-0004-2

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