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Spectral properties of disordered fully connected graphs.

S N Taraskin1

  • 1St. Catharine's College and Department of Chemistry, University of Cambridge, Cambridge, United Kingdom. snt1000@cam.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 31, 2005
PubMed
Summary

This study analyzes spectral properties of disordered graphs. Researchers derived an analytical expression for spectral density, enabling lower-bound estimates for critical parameters in contact processes.

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Area of Science:

  • Graph theory
  • Statistical physics
  • Condensed matter physics

Background:

  • Disordered fully connected graphs exhibit complex spectral properties.
  • Node-node interactions influence these properties in electronic and vibrational systems.
  • Understanding these properties is crucial for modeling phenomena like contact processes.

Purpose of the Study:

  • To investigate the spectral properties of disordered fully connected graphs.
  • To derive an analytical expression for ensemble-averaged spectral density.
  • To establish methods for estimating critical parameters in related processes.

Main Methods:

  • Derivation of an approximate analytical expression for spectral density.
  • Analysis of the Hamiltonian for electronic and vibrational problems.

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  • Evaluation of extreme eigenvalues for bound estimation.

    • Main Results:

    • An approximate analytical expression for spectral density was derived.
    • The derived expression is applicable to both electronic and vibrational problems.
    • A method for estimating lower bounds of critical parameters was demonstrated.

    Conclusions:

    • The spectral properties of disordered graphs can be analytically approximated.
    • Extreme eigenvalue evaluation provides insights into critical parameters.
    • This work offers a framework for studying complex systems on disordered graphs.