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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
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Characteristic polynomials, spectral-based Riemann-Zeta functions and entropy indices of n-dimensional hypercubes.

Krishnan Balasubramanian1

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|June 26, 2023
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Summary

Researchers computed spectral indices for n-dimensional hypercubes using recursive Hadamard transforms. Graph energies showed a J-curve, while spectral entropies linearly depended on dimension.

Keywords:
Characteristic polynomialsRiemann-Zeta functions of hypercubesSpectra of hypercubesSpectral-based entropiesnD-hypercubes

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Area of Science:

  • Graph theory
  • Spectral analysis
  • Computational mathematics

Background:

  • n-dimensional hypercubes are fundamental structures in graph theory.
  • Spectral analysis provides insights into graph properties.
  • Understanding hypercube spectra is crucial for various applications.

Purpose of the Study:

  • To compute characteristic polynomials and spectral indices of n-dimensional hypercubes.
  • To analyze the behavior of graph energies and spectral entropies with increasing dimension.
  • To provide structural interpretations for polynomial coefficients and derive expressions for spectral functions.

Main Methods:

  • Utilizing recursive Hadamard transforms for spectral computations.
  • Calculating Riemann-Zeta functional indices and spectral entropies.
  • Analyzing numerical results for hypercubes up to 23 dimensions.

Main Results:

  • Obtained characteristic polynomials and spectral indices for n-dimensional hypercubes.
  • Observed a J-curve relationship between graph energies and hypercube dimension.
  • Identified a linear dependence of spectral entropies on hypercube dimension.
  • Provided structural interpretations for coefficients of characteristic polynomials.

Conclusions:

  • Recursive Hadamard transforms are effective for spectral analysis of hypercubes.
  • The dimension of n-cubes significantly influences graph energies and spectral entropies.
  • Derived new expressions for spectral-based Riemann-Zeta functions and their associated integer sequences.