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<p>Let <i>K</i> be an <i>N</i>-dimensional simplicial complex. We investigate the spectrum of the up Laplacian matrix of <i>K</i>. Let <i>L</i> be the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2146_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\((N-1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>th up Laplacian matrix of <i>K</i>. We show that the largest eigenvalues of <i>L</i> and |<i>L</i>| are equal if and only if <i>K</i> is disorientable. We also derive lower bounds for the sum of the first <i>k</i> largest eigenvalues of <i>L</i>.</p>

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On Laplacians and the disorientability of a simplicial complex

  • 期刊论文

  • 发表日期：

    2025年7月9日

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  • R. Balaji,
  • Gargi Lather,
  • Vinayak Gupta

摘要

Let K be an N-dimensional simplicial complex. We investigate the spectrum of the up Laplacian matrix of K. Let L be the \((N-1)\) ( N - 1 ) th up Laplacian matrix of K. We show that the largest eigenvalues of L and |L| are equal if and only if K is disorientable. We also derive lower bounds for the sum of the first k largest eigenvalues of L.