{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,10]],"date-time":"2023-06-10T04:10:19Z","timestamp":1686370219561},"reference-count":18,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["era"],"published-print":{"date-parts":[[2022]]},"abstract":"<jats:p xml:lang=\"fr\">&lt;abstract&gt;&lt;p&gt;In this note, we approximate the von Neumann and R\u00e9nyi 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.&lt;\/p&gt;&lt;\/abstract&gt;<\/jats:p>","DOI":"10.3934\/era.2022094","type":"journal-article","created":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T03:01:35Z","timestamp":1648782095000},"page":"1864-1880","source":"Crossref","is-referenced-by-count":0,"title":["Approximations for the von Neumann and R\u00e9nyi entropies of graphs with circulant type Laplacians"],"prefix":"10.3934","volume":"30","author":[{"given":"Nat\u00e1lia","family":"Bebiano","sequence":"first","affiliation":[{"name":"CMUC, Department of Mathematics, University of Coimbra, P 3001-454 Coimbra, Portugal"}]},{"given":"Jo\u00e3o","family":"da Provid\u00eancia","sequence":"additional","affiliation":[{"name":"CFisUC, Department of Physics, University of Coimbra, P 3004-516 Coimbra, Portugal"}]},{"given":"Wei-Ru","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Laurent Mathematics Center, Sichuan Normal University, Chengdu 610066, China"}]}],"member":"2321","reference":[{"key":"key-10.3934\/era.2022094-1","doi-asserted-by":"publisher","unstructured":"N. Bebiano, S. Furtado, J. da Provid\u00eancia, W. R. Xu, J. P. da Provid\u00eancia, Approximations for the von Neumann and Renyi entropies of graphs using the Euler-Maclaurin formula, <i>Electron. Trans. Numer. Anal.<\/i>, <b>48<\/b> (2018), 227\u2013242. https:\/\/doi.org\/10.1553\/etna_vol48s227","DOI":"10.1553\/etna_vol48s227"},{"key":"key-10.3934\/era.2022094-2","doi-asserted-by":"publisher","unstructured":"M. Dairyko, L. Hogben, J. C. H. Lin, J. Lockhart, D. Roberson, S. Severini, et al., Note on von Neumann and R\u00e9nyi entropies of a graph, <i>Linear Algebra Appl.<\/i>, <b>521<\/b> (2017), 240\u2013253. https:\/\/doi.org\/10.1016\/j.laa.2017.01.037","DOI":"10.1016\/j.laa.2017.01.037"},{"key":"key-10.3934\/era.2022094-3","unstructured":"V. Giovannetti, S. Severini, The Kirchhoff's matrix tree theorem revisited: counting spanning trees with the quantum relative entropy, preprint, arXiv: 1102.2398. <a href=\"https:\/\/doi.org\/10.48550\/arXiv.1102.2398\" target=\"_blank\">https:\/\/doi.org\/10.48550\/arXiv.1102.2398<\/a>"},{"key":"key-10.3934\/era.2022094-4","doi-asserted-by":"publisher","unstructured":"H. Lin, B. Zhou, On the von Neumann entropy of a graph, <i>Discrete Appl. Math.<\/i>, <b>247<\/b> (2018), 448\u2013455. https:\/\/doi.org\/10.1016\/j.dam.2018.04.004","DOI":"10.1016\/j.dam.2018.04.004"},{"key":"key-10.3934\/era.2022094-5","doi-asserted-by":"publisher","unstructured":"G. Minello, L. Rossi, A. Torsello, On the von Neumann entropy of graphs, <i>J. Complex Networks<\/i>, <b>7<\/b> (2019), 491\u2013514. https:\/\/doi.org\/10.1093\/comnet\/cny028","DOI":"10.1093\/comnet\/cny028"},{"key":"key-10.3934\/era.2022094-6","doi-asserted-by":"publisher","unstructured":"D. E. Simmons, J. P. Coon, A. Datta, Symmetric Laplacians, quantum density matrices and their von Neumann entropy, <i>Linear Algebra Appl.<\/i>, <b>521<\/b> (2017), 240\u2013253. https:\/\/doi.org\/10.1016\/j.laa.2017.06.038","DOI":"10.1016\/j.laa.2017.06.038"},{"key":"key-10.3934\/era.2022094-7","doi-asserted-by":"publisher","unstructured":"D. E. Simmons, J. P. Coon, A. Datta, The von Neumann Theil index: characterizing graph centralization using the von Neumann index, <i>J. Complex Networks<\/i>, <b>6<\/b> (2018), 859\u2013876. https:\/\/doi.org\/10.1093\/comnet\/cnx061","DOI":"10.1093\/comnet\/cnx061"},{"key":"key-10.3934\/era.2022094-8","doi-asserted-by":"publisher","unstructured":"C. Ye, R. C. Wilson, C. H. Comin, L. F. Costa, E. R. Hancock, Approximate von Neumann entropy for direct graphs, <i>Phys. Rev. E<\/i>, <b>89<\/b> (2014), 052804. https:\/\/doi.org\/10.1103\/PhysRevE.89.052804","DOI":"10.1103\/PhysRevE.89.052804"},{"key":"key-10.3934\/era.2022094-9","doi-asserted-by":"publisher","unstructured":"R. Grone, R. Merris, V. S. Sunder, The Laplacian spectrum of a graph, <i>SIAM J. Matrix Anal. Appl.<\/i>, <b>11<\/b> (1990), 218\u2013238. https:\/\/doi.org\/10.1137\/0611016","DOI":"10.1137\/0611016"},{"key":"key-10.3934\/era.2022094-10","unstructured":"R. A. Horn, C. R. Johnson, <i>Matrix Analysis<\/i>, 2<sup><i>nd<\/i><\/sup> edition, Cambridge University Press, Cambridge, 2013."},{"key":"key-10.3934\/era.2022094-11","unstructured":"J. V. Neumann, <i>Mathematical Foundations of Quantum Mechanics, Number 2<\/i>, Princeton University Press, Princeton, N.J., 1955."},{"key":"key-10.3934\/era.2022094-12","unstructured":"L. D. Landau, E. M. Lifshitz, <i>Statistical Physics<\/i>, Pergamon Press, Oxford-Edinburgh-New York, 1969."},{"key":"key-10.3934\/era.2022094-13","unstructured":"A. R\u00e9nyi, <i>On measures of entropy and information<\/i>, in <i>Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics<\/i>, <b>4<\/b> (1961), 547\u2013562."},{"key":"key-10.3934\/era.2022094-14","doi-asserted-by":"publisher","unstructured":"T. M. Apostol, An elementary view of Euler's summation formula, <i>Amer. Math. Monthly<\/i>, <b>106<\/b> (1999), 409\u2013418. https:\/\/doi.org\/10.1080\/00029890.1999.12005063","DOI":"10.1080\/00029890.1999.12005063"},{"key":"key-10.3934\/era.2022094-15","doi-asserted-by":"publisher","unstructured":"V. Lampret, The Euler\u2013Maclaurin and Taylor formulas: twin, elementary derivations, <i>Math. Mag.<\/i>, <b>74<\/b> (2001), 109\u2013122. https:\/\/doi.org\/10.1080\/0025570X.2001.11953046","DOI":"10.1080\/0025570X.2001.11953046"},{"key":"key-10.3934\/era.2022094-16","unstructured":"V. Lampret, The Euler-Maclaurin Formula and Sums of Powers Revisited, <i>Int. J. Contemp. Math. Sci.<\/i>, <b>5<\/b> (2010), 2401\u20132407."},{"key":"key-10.3934\/era.2022094-17","doi-asserted-by":"publisher","unstructured":"M. Z. Spivey, The Euler-Maclaurin formula and sums of powers, <i>Math. Mag.<\/i>, <b>79<\/b> (2006), 61\u201365. https:\/\/doi.org\/10.1080\/0025570X.2006.11953378","DOI":"10.1080\/0025570X.2006.11953378"},{"key":"key-10.3934\/era.2022094-18","doi-asserted-by":"publisher","unstructured":"A. Greenbaum, R. C. Li, M. L. Overton, First-order perturbation theory for eigenvalues and eigenvectors, <i>SIAM Rev.<\/i>, <b>62<\/b> (2020), 463\u2013482. https:\/\/doi.org\/10.1137\/19M124784X","DOI":"10.1137\/19M124784X"}],"container-title":["Electronic Research Archive"],"original-title":[],"link":[{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/era.2022094?viewType=html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T03:26:24Z","timestamp":1686281184000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/era.2022094"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022]]},"references-count":18,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2022]]}},"URL":"https:\/\/doi.org\/10.3934\/era.2022094","relation":{},"ISSN":["2688-1594"],"issn-type":[{"value":"2688-1594","type":"print"}],"subject":[],"published":{"date-parts":[[2022]]}}}