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A 1-separation formula for the graph Kemeny constant and Braess edges

Abstract

Kemeny’s constant of a simple connected graph G is the expected length of a random walk from i to any given vertex \(j \ne i\). We provide a simple method for computing Kemeny’s constant for 1-separable graphs via effective resistance methods from electrical network theory. Using this formula, we furnish a simple proof that the path graph on n vertices maximizes Kemeny’s constant for the class of undirected trees on n vertices. Applying this method again, we simplify existing expressions for the Kemeny’s constant of barbell graphs and demonstrate which barbell maximizes Kemeny’s constant. This 1-separation identity further allows us to create sufficient conditions for the existence of Braess edges in 1-separable graphs. We generalize the notion of the Braess edge to Braess sets, collections of non-edges in a graph such that their addition to the base graph increases the Kemeny constant. We characterize Braess sets in graphs with any number of twin pendant vertices, generalizing work of Kirkland and Zeng (Electron J Linear Algebra 31(1):444–464, 2016) and Ciardo (Linear Algebra Appl, 2020).

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References

  1. 1.

    R.B. Bapat, Graphs and Matrices, vol. 27 (Springer, 2010)

  2. 2.

    W. Barrett, E.J. Evans, A.E. Francis, Resistance distance in straight linear 2-trees. Discrete Appl. Math. 258, 13–34 (2019)

    Article  Google Scholar 

  3. 3.

    D. Braess, Über ein paradoxon aus der verkehrsplanung. Unternehmensforschung 12(1), 258–268 (1968)

    Google Scholar 

  4. 4.

    D. Braess, A. Nagurney, T. Wakolbinger, On a paradox of traffic planning. Transp Sci. 39(4), 446–450 (2005)

    Article  Google Scholar 

  5. 5.

    J. Breen, S. Butler, N. Day, C. DeArmond, K. Lorenzen, H. Qian, J. Riesen, Computing Kemeny’s constant for a barbell graph. Electron J Linear Algebra 35, 583–598 (2019)

  6. 6.

    M. Catral, S.J. Kirkland, M. Neumann, N.S. Sze, The Kemeny constant for finite homogeneous ergodic Markov chains. J. Sci. Comput. 45(1), 151–166 (2010)

    Article  Google Scholar 

  7. 7.

    L. Ciardo, The Braess’ paradox for pendant twins. Linear Algebra Appl. 590, 304–316 (2020)

    Article  Google Scholar 

  8. 8.

    L. Ciardo, G. Dahl, S. Kirkland, On Kemeny’s constant for trees with fixed order and diameter. Linear Multilinear Algebra 1–23 (2020)

  9. 9.

    E. Estrada, N. Hatano, Topological atomic displacements, Kirchhoff and Wiener indices of molecules. Chem. Phys. Lett. 486(4–6), 166–170 (2010)

    CAS  Article  Google Scholar 

  10. 10.

    S. Hayat, S. Khan, M. Imran, J.-B. Liu, Quality testing of distance-based molecular descriptors for benzenoid hydrocarbons. J. Mol. Struct. 1222, 128927 (2020)

    CAS  Article  Google Scholar 

  11. 11.

    S. Kirkland, Z. Zeng, Kemeny’s constant and an analogue of Braess’ paradox for trees. Electron. J. Linear Algebra 31(1), 444–464 (2016)

  12. 12.

    D.J. Klein, M. Randić, Resistance distance. J. Math. Chem. 12(1), 81–95 (1993)

    Article  Google Scholar 

  13. 13.

    M. Levene, G. Loizou, Kemeny’s constant and the random surfer. Am. Math. Month. 109(8), 741–745 (2002)

  14. 14.

    M. Li, J. Xie, D. Lian, C.-F. Yang, Multiplicative degree-Kirchhoff index of random polyphenyl chains. Sens. Mater. 33(8), 2629–2638 (2021)

    Google Scholar 

  15. 15.

    S. Li, W. Sun, S. Wang, Multiplicative degree-Kirchhoff index and number of spanning trees of a zigzag polyhex nanotube TUHC [2\(n\), 2]. Int. J. Quantum Chem. 119(17), e25969 (2019)

    Article  Google Scholar 

  16. 16.

    Shuchao Li, Wei Wei, Yu. Shiqun, On normalized Laplacians, multiplicative degree-Kirchhoff indices, and spanning trees of the linear [\(n\)] phenylenes and their dicyclobutadieno derivatives. Int. J. Quantum Chem. 119(8), e25969 (2019)

    Article  Google Scholar 

  17. 17.

    J.L. Palacios, J.M. Renom, Broder and Karlin’s formula for hitting times and the Kirchhoff index. Int. J. Quantum Chem. 111(1), 35–39 (2011)

  18. 18.

    J.L. Palacios, J.M. Renom, Bounds for the Kirchhoff index of regular graphs via the spectra of their random walks. Int. J. Quantum Chem. 110(9), 1637–1641 (2010)

    CAS  Article  Google Scholar 

  19. 19.

    Y. Pan, J. Li, Kirchhoff index, multiplicative degree-Kirchhoff index and spanning trees of the linear crossed hexagonal chains. Int. J. Quantum Chem. 118(24), e25787 (2018)

    Article  Google Scholar 

  20. 20.

    R. Patel, P. Agharkar, F. Bullo, Robotic surveillance and Markov chains with minimal weighted Kemeny constant. IEEE Trans. Autom. Control 60(12), 3156–3167 (2015)

    Article  Google Scholar 

  21. 21.

    H. Wiener, Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69(1), 17–20 (1947)

    CAS  Article  Google Scholar 

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Correspondence to Mark Kempton.

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Faught, N., Kempton, M. & Knudson, A. A 1-separation formula for the graph Kemeny constant and Braess edges. J Math Chem (2021). https://doi.org/10.1007/s10910-021-01294-8

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Keywords

  • Kemeny’s constant
  • Effective resistance
  • Resistance distance
  • Graph theory
  • Graph connectivity