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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 792^{1} 8^{1035} (–36)^{252}.
Its complement has parameters
(*v*, *k*, λ, μ) = (1288, 495, 206, 180)
and spectrum 495^{1} 35^{252} (–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 M_{24}
acting rank 3 with point stabilizer M_{12}: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 1288^{1} 8^{1771} (–56)^{276}.
Its complement has parameters
(*v*, *k*, λ, μ) = (2048, 759, 310, 264)
and spectrum 759^{1} 55^{276} (–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 2^{11} cosets of {**0**,**1**} in C,
and join two cosets when they have distance 12.
## Group

The automorphism group of Σ is 2^{11}.M_{24}
acting rank 3 with point stabilizer M_{24}.
## 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.