%0 Journal Article
%T The Laplacian Permanents and Laplacian Ratios of Trees
%A Xiangshuai Dong
%J Applied Mathematics
%P 338-346
%@ 2152-7393
%D 2025
%I Scientific Research Publishing
%R 10.4236/am.2025.164017
%X Brualdi and Goldwasser characterized the Laplacian permanents of trees. In this paper, we study the Laplacian permanents of trees. We characterize some Laplacian permanents of trees. The Laplacian ratio of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>G</mi>
</math>

 is the Laplacian permanent of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>G</mi>
</math>

 divided by the product of degrees of all vertices. In this paper, we obtain that for any <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>n</mi>
</math>

-vertex caterpillar tree <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>T</mi>
</math>

, there exists an <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>n</mi>
</math>

-vertex caterpillar tree <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>C</mi>
   <mi>n</mi>
  </msub>
  </mrow>
</math>

 such that <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>&#x03C0;</mi><mrow><mo>(</mo>
   <mrow>
    <msub>
     <mi>C</mi>
     <mi>n</mi>
    </msub>
    </mrow>
  <mo>)</mo></mrow><mo>&#x2264;</mo><mi>&#x03C0;</mi><mrow><mo>(</mo>
   <mi>T</mi>
  <mo>)</mo></mrow></mrow>
</math>

.
%K Laplacian Permanents
%K Laplacian Ratio
%K Tree
%K Caterpillar
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=141946