Foster graph

<h2 id="algebraic-properties">Algebraic properties</h2>
The automorphism group of the Foster graph is a group of order 4320.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Foster graph is a <a href="/facts/Symmetric_graph/vPVWrnjb">symmetric graph</a>. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the Foster graph, referenced as F90A, is the only cubic symmetric graph on 90 vertices.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
The <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> of the Foster graph is equal to 
 
 
 
 (
 x
 −
 3
 )
 (
 x
 −
 2
 
 )
 
 9
 
 
 (
 x
 −
 1
 
 )
 
 18
 
 
 
 x
 
 10
 
 
 (
 x
 +
 1
 
 )
 
 18
 
 
 (
 x
 +
 2
 
 )
 
 9
 
 
 (
 x
 +
 3
 )
 (
 
 x
 
 2
 
 
 −
 6
 
 )
 
 12
 
 
 
 
 {\displaystyle (x-3)(x-2)^{9}(x-1)^{18}x^{10}(x+1)^{18}(x+2)^{9}(x+3)(x^{2}-6)^{12}}
 
.

<h2 id="gallery">Gallery</h2>

<ul><li>Biggs, N. L.; Boshier, A. G.; Shawe-Taylor, J. (1986), "Cubic distance-regular graphs", Journal of the London Mathematical Society, 33 (3): 385–394, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fjlms%2Fs2-33.3.385">10.1112/jlms/s2-33.3.385</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0850954">0850954</a>.</li></ul>
<ul><li>Van Dam, Edwin R.; Haemers, Willem H. (2002), "Spectral characterizations of some distance-regular graphs", Journal of Algebraic Combinatorics, 15 (2): 189–202, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1023%2FA%3A1013847004932">10.1023/A:1013847004932</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1887234">1887234</a>.</li></ul>
<ul><li>Van Maldeghem, Hendrik (2002), "Ten exceptional geometries from trivalent distance regular graphs", <a href="/facts/Annals_of_Combinatorics/2v3tCE7a">Annals of Combinatorics</a>, 6 (2): 209–228, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FPL00012587">10.1007/PL00012587</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1955521">1955521</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:195315348">195315348</a>.</li></ul>

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

<ol>
<li id="fn:1">Weisstein, Eric W. "Foster Graph". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Cubic distance-regular graphs, A. Brouwer. <a href="http://www.win.tue.nl/~aeb/graphs/cubic_drg.html" target="_blank">http://www.win.tue.nl/~aeb/graphs/cubic_drg.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Hiraki, Akira; Nomura, Kazumasa; Suzuki, Hiroshi (2000), "Distance-regular graphs of valency 6 and 
 
 
 
 
 a
 
 1
 
 
 =
 1
 
 
 {\displaystyle a_{1}=1}
 
", Journal of Algebraic Combinatorics, 11 (2): 101–134, doi:10.1023/A:1008776031839, MR 1761910 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">"G-12 Foster graph", Encyclopedia of Graphs, retrieved 2024-02-26 <a href="http://atlas.gregas.eu/graphs/33" target="_blank">http://atlas.gregas.eu/graphs/33</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002. <a href="/wiki/Marston_Conder" target="_blank">/wiki/Marston_Conder</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Foster graph open-in-new

Foster graph