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The dodecad graph
There is a strongly regular graph Γ with parameters
(v, k, λ, μ) = (1288, 792, 476, 504).
The spectrum is 7921 81035 (–36)252.
Its complement has parameters
(v, k, λ, μ) = (1288, 495, 206, 180)
and spectrum 4951 35252 (–9)1035.
Construction is folklore. The graph is for example given in
Hubaut (1975). Uniqueness is unknown.
Construction
Let C be the extended binary Golay code.
It has 2576 words of weight 12 (dodecads), so 1288 complementary pairs
of dodecads. Given one dodecad, there are 1, 495, 1584, 495, 1 dodecads
at distance 0, 8, 12, 16, 24, respectively.
Given one complementary pair of dodecads, there are 1, 495, 792
such pairs at distance 0, 8, 12, respectively.
The graph Γ is obtained if we call two dodecad pairs adjacent if
they have distance 12.
Group
The automorphism group of Γ is M24
acting rank 3 with point stabilizer M12:2.
Supergraphs
This graph is the local graph of a strongly regular graph Σ
with parameters
(v, k, λ, μ) = (2048, 1288, 792, 840)
and spectrum 12881 81771 (–56)276.
Its complement has parameters
(v, k, λ, μ) = (2048, 759, 310, 264)
and spectrum 7591 55276 (–9)1771.
The graph Σ was constructed by Goethals and Seidel (1970),
who write `we have reasons to believe that the subgraph on the 1288
neighbours of any vertex is strongly regular'.
Uniqueness is unknown.
Construction
Let C be the extended binary Golay code. It has
1, 759, 2576, 759, 1 words of weight 0, 8, 12, 16, 24, respectively.
Take the 211 cosets of {0,1} in C,
and join two cosets when they have distance 12.
Group
The automorphism group of Σ is 211.M24
acting rank 3 with point stabilizer M24.
References
J. M. Goethals & J. J. Seidel,
Strongly regular graphs derived from combinatorial designs,
Canad. J. Math. 22 (1970) 597-614.
X. L. Hubaut,
Strongly regular graphs,
Discr. Math. 13 (1975) 357-381.