Levi graph

<h2 id="examples">Examples</h2>

<ul><li>The <a href="/facts/Desargues_graph/M6sXWfFK">Desargues graph</a> is the Levi graph of the <a href="/facts/Desargues_configuration/Y3wdewhj">Desargues configuration</a>, composed of 10 points and  10 lines. There are 3 points on each line, and 3 lines passing through each point. The Desargues graph can also be viewed as the <a href="/facts/Generalized_Petersen_graph/dnmt90Ww">generalized Petersen graph</a> G(10,3) or the <a href="/facts/Kneser_graph/Iu0TVMKT">bipartite Kneser graph</a> with parameters 5,2. It is 3-regular with 20 vertices.</li>
<li>The <a href="/facts/Heawood_graph/ymqQSkIK">Heawood graph</a> is the Levi graph of the <a href="/facts/Fano_plane/nNuh1gkH">Fano plane</a>. It is also known as the (3,6)-<a href="/facts/Cage_(graph_theory)/Ya4RhwvU">cage</a>, and is 3-regular with 14 vertices.</li>
<li>The <a href="/facts/M%25C3%25B6bius%25E2%2580%2593Kantor_graph/T7NyYuxL">Möbius–Kantor graph</a> is the Levi graph of the <a href="/facts/M%25C3%25B6bius%25E2%2580%2593Kantor_configuration/NgQIafId">Möbius–Kantor configuration</a>, a system of 8 points and 8 lines that cannot be realized by straight lines in the Euclidean plane. It is 3-regular with 16 vertices.</li>
<li>The <a href="/facts/Pappus_graph/w40dHpA6">Pappus graph</a> is the Levi graph of the <a href="/facts/Pappus_configuration/YYsP7euh">Pappus configuration</a>, composed of 9 points and 9 lines. Like the Desargues configuration there are 3 points on each line and 3 lines passing through each point. It is 3-regular with 18 vertices.</li>
<li>The <a href="/facts/Gray_graph/Xgfh4m3X">Gray graph</a> is the Levi graph of a configuration that can be realized in 
  
    
      
        
          
            R
          
          
            3
          
        
      
    
    {\displaystyle \mathbb {R} ^{3}}
  
 as a 
  
    
      
        3
        ×
        3
        ×
        3
      
    
    {\displaystyle 3\times 3\times 3}
  
 grid of 27 points and the 27 orthogonal lines through them.</li>
<li>The <a href="/facts/Tutte_eight-cage/SH13hLkX">Tutte eight-cage</a> is the Levi graph of the <a href="/facts/Cremona%25E2%2580%2593Richmond_configuration/14F45Gca">Cremona–Richmond configuration</a>. It is also known as the (3,8)-cage, and is 3-regular with 30 vertices.</li>
<li>The four-dimensional <a href="/facts/Hypercube_graph/pwfGEl9J">hypercube graph</a> 
  
    
      
        
          Q
          
            4
          
        
      
    
    {\displaystyle Q_{4}}
  
 is the Levi graph of the <a href="/facts/M%25C3%25B6bius_configuration/Pyukr6Wc">Möbius configuration</a> formed by the points and planes of two mutually incident tetrahedra.</li>
<li>The <a href="/facts/Ljubljana_graph/YIjevVKE">Ljubljana graph</a> on 112 vertices is the Levi graph of the Ljubljana configuration.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/LeviGraph.html">"Levi Graph"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Grünbaum, Branko (2006). "Configurations of points and lines". The Coxeter Legacy. Providence, RI: American Mathematical Society. pp. 179–225. MR 2209028.. See in particular p. 181. <a href="/wiki/Branko_Gr%C3%BCnbaum" target="_blank">/wiki/Branko_Gr%C3%BCnbaum</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Polster, Burkard (1998). A Geometrical Picture Book. Universitext. New York: Springer-Verlag. p. 5. doi:10.1007/978-1-4419-8526-2. ISBN 0-387-98437-2. MR 1640615.. <a href="0-387-98437-2" target="_blank">0-387-98437-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Grünbaum, Branko (2006). "Configurations of points and lines". The Coxeter Legacy. Providence, RI: American Mathematical Society. pp. 179–225. MR 2209028.. See in particular p. 181. <a href="/wiki/Branko_Gr%C3%BCnbaum" target="_blank">/wiki/Branko_Gr%C3%BCnbaum</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Levi, F. W. (1942). Finite Geometrical Systems. Calcutta: University of Calcutta. MR 0006834.. <a href="/wiki/Friedrich_Wilhelm_Levi" target="_blank">/wiki/Friedrich_Wilhelm_Levi</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Grünbaum, Branko (2006). "Configurations of points and lines". The Coxeter Legacy. Providence, RI: American Mathematical Society. pp. 179–225. MR 2209028.. See in particular p. 181. <a href="/wiki/Branko_Gr%C3%BCnbaum" target="_blank">/wiki/Branko_Gr%C3%BCnbaum</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Gropp, Harald (2007). "VI.7 Configurations". In Colbourn, Charles J.; Dinitz, Jeffrey H. (eds.). Handbook of combinatorial designs. Discrete Mathematics and its Applications (Boca Raton) (Second ed.). Chapman & Hall/CRC, Boca Raton, Florida. pp. 353–355.. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Conder, Marston; Malnič, Aleksander; Marušič, Dragan; Pisanski, Tomaž; Potočnik, Primož (2002). The Ljubljana Graph (PDF) (IMFM Preprint). Vol. 40–845. University of Ljubljana Department of Mathematics.. <a href="/wiki/Marston_Conder" target="_blank">/wiki/Marston_Conder</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Levi graph open-in-new

Levi graph