%0 Journal Article
%T The Seidel Eigenvalues Polynomial and Spectrum of the Petersen Graph
%A Dan Zhou
%A Shengmei Lv
%J Open Journal of Applied Sciences
%P 1025-1032
%@ 2165-3925
%D 2025
%I Scientific Research Publishing
%R 10.4236/ojapps.2025.154071
%X For a simple undirected graph <i>G</i>, let <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>A</mi><mrow><mo>(</mo>
   <mi>G</mi>
  <mo>)</mo></mrow></mrow>
</math>

 be the (0, 1) adjacency matrix of <i>G</i>. The Seidel matrix of <i>G</i>, is defined as <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>S</mi><mrow><mo>(</mo>
   <mi>G</mi>
  <mo>)</mo></mrow><mo>=</mo><mi>J</mi><mo>&#x2212;</mo><mi>I</mi><mo>&#x2212;</mo><mn>2</mn><mi>A</mi><mrow><mo>(</mo>
   <mi>G</mi>
  <mo>)</mo></mrow></mrow>
</math>

, where <i>J</i> is the all-one matrix and <i>I</i> is the identity matrix. The Seidel eigenvalues polynomial of the graph <i>G</i> is <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>S</mi>
   <mi>G</mi>
  </msub>
  <mrow><mo>(</mo>
   <mi>&#x03BB;</mi>
  <mo>)</mo></mrow><mo>=</mo><mi>det</mi><mrow><mo>(</mo>
   <mrow>
    <mi>&#x03BB;</mi><mi>I</mi><mo>&#x2212;</mo><mi>S</mi><mrow><mo>(</mo>
     <mi>G</mi>
    <mo>)</mo></mrow></mrow>
  <mo>)</mo></mrow></mrow>
</math>

. If all the Seidel eigenvalues of the graph <i>G</i> are integers, then <i>G</i> is called a Seidel integer graph. In this paper, we apply methods from algebraic and matrix theory to obtain the Seidel eigenvalue polynomials of the Petersen graph. Furthermore, we determine whether the Petersen graph is a Seidel integral graph.
%K Petersen Graph
%K Seidel Eigenvalues Polynomial
%K Seidel Spectral
%K Seidel Integral Graph
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=142012