Rocky Mountain Journal of Mathematics

Lower bounds for the Estrada index using mixing time and Laplacian spectrum

Yilun Shang

Full-text: Open access


The logarithm of the Estrada index has been recently proposed as a spectral measure to characterize the robustness of complex networks. We derive novel analytic lower bounds for the logarithm of the Estrada index based on the Laplacian spectrum and the mixing times of random walks on the network. The main techniques employed are some inequalities, such as the thermodynamic inequality in statistical mechanics, a trace inequality of von Neumann, and a refined harmonic-arithmetic mean inequality.

Article information

Rocky Mountain J. Math., Volume 43, Number 6 (2013), 2009-2016.

First available in Project Euclid: 25 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A42: Inequalities involving eigenvalues and eigenvectors 05C81: Random walks on graphs

Estrada index mixing time Laplacian matrix random walk natural connectivity


Shang, Yilun. Lower bounds for the Estrada index using mixing time and Laplacian spectrum. Rocky Mountain J. Math. 43 (2013), no. 6, 2009--2016. doi:10.1216/RMJ-2013-43-6-2009.

Export citation


  • N. Bebiano, J. da Providência, Jr. and R. Lemos, Matrix inequalities in statistical mechanics, Lin. Alg. Appl. 376 (2004), 265-273.
  • N. Biggs, Algebraic graph theory, Cambridge University Press, Cambridge, 1993.
  • M. Catral, S.J. Kirkland, M. Neumann and N.-S. Sze, The Kemeny constant for finite homogeneous ergodic Markov chains, J. Sci. Comput. 45 (2010), 151-166.
  • F. Chung, Spectral graph theory, American Mathematical Society, Providence, 1997.
  • V. Chvátal, Tough graphs and Hamiltonian circuits, Discrete Math. 5 (1973), 215-228.
  • E. Estrada, Characterization of $3$D molecular structure, Chem. Phys. Lett. 319 (2000), 713-718.
  • E. Estrada and N. Hatano, Statistical-mechanical approach to subgraph centrality in complex networks, Chem. Phys. Lett. 439 (2007), 247-251.
  • E. Estrada and J.A. Rodríguez-Velázquez, Spectral measures of bipartivity in complex networks, Phys. Rev. 72 (2005), 046105.
  • –––, Subgraph centrality in complex networks, Phys. Rev. 71 (2005), 056103.
  • M. Fiedler, Algebraic connectivity of graphs, Czech. Math. J. 23 (1973), 298-305.
  • F. Harary, Conditional connectivity, Networks 13 (1983), 346-357.
  • K. Huang, Statistical mechanics, John Wiley & Sons, New York, 1987.
  • J.J. Hunter, Mixing times with applications to perturbed Markov chains, Lin. Alg. Appl. 417 (2006), 108-123.
  • H. Jung, On a class of posets and the corresponding comparability graphs, J. Comb. Theory 24 (1978), 125-133.
  • S. Kirkland, Fastest expected time to mixing for a Markov chain on a directed graph, Lin. Alg. Appl. 433 (2010), 1988-1996.
  • M. Krishnamoorth and B. Krishnamirthy, Fault diameter of interconnection networks, Comput. Math. Appl. 13 (1987), 577-582.
  • M. Levene and G. Loizou, Kemeny's constant and the random surfer, Amer. Math. Month. 109 (2002), 741-745.
  • L. Lovász, Random walks on graphs: A survey, in Combinatorics, Paul Erdos is Eighty, Bolyai Soc. Math. Stud. 2 (1993), 1-46.
  • P.R. Mercer, Refined arithmetic, geometric and harmonic mean inequalities, Rocky Mountain J. Math. 33 (2003), 1459-1464.
  • L. Mirsky, A trace inequality of John von Neumann, Monat. Math. 79 (1975), 303-306.
  • B. Mohar, The Laplacian spectrum of graphs, in Graph theory, combinatorics, and applications, Vol. 2, Wiley, 1991.
  • –––, Isoperimetric numbers of graphs, J. Comb. Theory B 47 (1989), 274-291.
  • M.E.J. Newman, The structure and function of complex networks, SIAM Rev. 45 (2003), 167-256.
  • E. Seneta, Non-negative matrices and Markov chains, Springer, New York, 2006.
  • Y. Shang, Biased edge failure in scale-free networks based on natural connectivity, Indian J. Phys. 86 (2012), 485-488.
  • –––, Local natural connectivity in complex networks, Chin. Phys. Lett. 28 (2011), Art. No. 068903.
  • –––, Perturbation results for the Estrada index in weighted networks, J. Phys. Math. Theor. 44 (2011), Art. No. 075003.
  • –––, The Estrada index of random graphs, Sci. Magna 7 (2011), 79-81.
  • J. Wu, M. Barahona, Y.J. Tan and H.Z. Deng, Natural connectivity of complex networks, Chin. Phys. Lett. 27 (2010), Art. No. 078902. \noindentstyle