The 231-Cameron graph is a strongly regular Hamiltonian graph on 231 vertices with parameters
. It is
distance-regular with intersection
array 
,
but is not distance-transitive.
It is named after mathematician Peter J. Cameron (Cameron et al. 1978).
It can be constructed by taking as vertices the 
unordered pairs from the point set of the Steiner
triple system 
and joining two vertices when the pairs are disjoint and their union is contained
in a block (Brouwer and van Lint 1984).
It has graph spectrum 
, and is therefore an integral
graph. It has graph automorphism group
order 
.
It is a Hamiltonian graph.
The 231-Cameron graph is implemented in the Wolfram Language as GraphData["CameronGraph"].
A different set of graphs associated with the name Cameron is the set of cubic planar Hamiltonian
graphs having exactly 3 Hamiltonian cycles
that were constructed by Kathleen Cameron (Cameron 2001; Knuth 2025, Problem 77).
The order-
planar cubic Cameron graph has 
vertices. The first few are illustrated above.
Since they are cubic and contain exactly 3 Hamiltonian cycles, they are automtatically perfectly
Hamiltonian. The 
and 
cases are Halin graphs. For 
, the graphs have exactly two graph
automorphisms (Knuth 2025).
For even 
,
each of the three Hamiltonian cycles gives a
bilaterally symmetric LCF embedding. "Nice"
LCF embeddings are shown above for 
to 5.
These graphs will be implemented in a future version of the Wolfram Language as GraphData[
"Cameron", n]
.
See also
Distance-Regular Graph, Distance-Transitive Graph, Strongly Regular Graph
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References
Brouwer, A. E. "Cameron Graph." http://www.win.tue.nl/~aeb/drg/graphs/Cameron.html.Brouwer, A. E. and van Lint, J. H. "Strongly Regular Graphs and Partial Geometries." In Enumeration and Design: Papers from the conference on combinatorics held at the University of Waterloo, Waterloo, Ont., June 14-July 2, 1982 (Ed. D. M. Jackson and S. A. Vanstone). Toronto, Canada: Academic Press, pp. 85-122, 1984.Brouwer, A. E. and van Maldeghem, H. "The Cameron Graph." ยง10.54 in Strongly Regular Graphs. Cambridge, England: Cambridge University Press, pp. 332-333, 2022.Cameron, K. "Thomason's Algorithm for Finding a Second Hamiltonian Circuit Through a Given Edge in a Cubic Graph Is Exponential on Krawczyk's Graphs." Disc. Math. 235, 69-77, 2001.Cameron, P. J.; Goethals, J.-M.; and Seidel, J. J. "Strongly Regular Graphs having Strongly Regular Subconstituents." J. Alg. 55, 257-280, 1978.DistanceRegular.org. "Cameron Graph." https://www.math.mun.ca/distanceregular/graphs//cameron.html.Knuth, D. E. "Hamiltonian Paths and Cycles." Volume 4, Pre-Fascicle 8A in The Art of Computer Programming. Draft. Aug. 19, 2025. https://www-cs-faculty.stanford.edu/~knuth/fasc8a.ps.gz.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Cameron Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CameronGraph.html