The Tutte 8-cage (Godsil and Royle 2001, p. 59; right figure) is a cubic graph on 30 nodes and 45 edges which is the Levi graph
of the Cremona-Richmond configuration.
It consists of the union of the two leftmost subgraphs illustrated above. The Tutte
8-cage is the unique -cage graph and Moore graph.
It is also a generalized polygon which is
the point/line Levi graph of the generalized quadrangle
and its line
graph is the generalized octagon. The graph was first discovered
by Tutte (1947) and is also called the Tutte-Coxeter graph (Bondy and Murty 1976,
p. 237; Brouwer et al. 1989, p. 209) or Tutte's cage (Read and Wilson
1998, p. 271).
The Tutte 8-cage is illustrated above in a number of embeddings.
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K.; Haythorpe, M.; Newcombe, A.; and Pegg, E. Jr. "There Are No Cubic Graphs
on 26 Vertices with Crossing Number 10 or 11." Preprint. 2019.Conder,
M. and Stokes, K. "New Methods for Finding Minimum Genus Embeddings of Graphs
on Orientable and Non-Orientable Surfaces." Ars. Math. Contemp.17,
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"Twelve Points in PG(5,3) with 95040 Self-Transformations." Proc. Roy.
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G. "Rectilinear Drawings of Famous Graphs: The Tutte 8-Cage." http://isu.indstate.edu/ge/COMBIN/RECTILINEAR/tutte.gif.Frucht,
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at Work: A Collection of Papers Showing Applications of Geometry (Ed. C. A. Gorini).
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and Wilson, R. J. An
Atlas of Graphs. Oxford, England: Oxford University Press, p. 271, 1998.Royle, G. "F030A."
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Cages." http://school.maths.uwa.edu.au/~gordon/remote/cages/Tutte,
W. T. "A Family of Cubical Graphs." Proc. Cambridge Philos. Soc.,
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-cage graph and Moore graph.
It is also a generalized polygon which is
the point/line Levi graph of the generalized quadrangle
and its line
graph is the generalized octagon 
. The graph was first discovered
by Tutte (1947) and is also called the Tutte-Coxeter graph (Bondy and Murty 1976,
p. 237; Brouwer et al. 1989, p. 209) or Tutte's cage (Read and Wilson
1998, p. 271).
.
The Tutte 8-cage graph is an integral graph with
graph spectrum 
. It can be represented in LCF
notation as ![[-13,-9,7,-7,9,13]^5](/images/equations/Tutte8-Cage/Inline6.svg)
(Frucht 1976).
