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Approximations for the von Neumann and Rényi entropies of graphs with circulant type Laplacians

  • We dedicate this paper to the memory of João da Providência, who unfortunately passed away just before the paper was published. da Providência played a crucial role in this research and he will be sorely missed
  • Received: 15 December 2021 Revised: 21 March 2022 Accepted: 28 March 2022 Published: 31 March 2022
  • In this note, we approximate the von Neumann and Rényi entropies of high-dimensional graphs using the Euler-Maclaurin summation formula. The obtained estimations have a considerable degree of accuracy. The performed experiments suggest some entropy problems concerning graphs whose Laplacians are $ g $-circulant matrices, i.e., circulant matrices with $ g $-periodic diagonals, or quasi-Toeplitz matrices. Quasi means that in a Toeplitz matrix the first two elements in the main diagonal, and the last two, differ from the remaining diagonal entries by a perturbation.

    Citation: Natália Bebiano, João da Providência, Wei-Ru Xu. Approximations for the von Neumann and Rényi entropies of graphs with circulant type Laplacians[J]. Electronic Research Archive, 2022, 30(5): 1864-1880. doi: 10.3934/era.2022094

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  • In this note, we approximate the von Neumann and Rényi entropies of high-dimensional graphs using the Euler-Maclaurin summation formula. The obtained estimations have a considerable degree of accuracy. The performed experiments suggest some entropy problems concerning graphs whose Laplacians are $ g $-circulant matrices, i.e., circulant matrices with $ g $-periodic diagonals, or quasi-Toeplitz matrices. Quasi means that in a Toeplitz matrix the first two elements in the main diagonal, and the last two, differ from the remaining diagonal entries by a perturbation.



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