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Graph

There is a unique strongly regular graph Γ with parameters v = 275, k = 112, λ = 30, μ = 56. The spectrum is 1121 2252 (–28)22. The complementary graph has parameters v = 275, k = 162, λ = 105, μ = 81 and spectrum 1621 2722 (–3)252. Uniqueness is due to Goethals & Seidel (1975). For a detailed discussion see Brouwer & van Lint (1984).

Group

The full group of automorphisms of Γ is G = McL.2 (of order 28.36.53.7.11) acting rank 3, with point stabilizer U4(3):2.

Construction

A 22+77+176 construction is found by taking the Steiner system S(4,7,23) with 1+22 points and 253 = 77+176 blocks, where the first 77 are those containing the first point. Use p,B,C to denote one of the 22, 77, 176 objects, and let ~ denote adjacency. Make the 22 remaining points a coclique, let p ~ B when p is not in B, let B ~ B' when B,B' meet in 1 point, let p ~ C when p is in C, let B ~ C when B,C meet in 3 points, let C ~ C' when C,C' meet in 1 point. This yields Γ. The resulting partition is equitable with parameters (0,56,56), (16,16,80), (7,35,70).

The same construction in other words: Take the 23 points and 253 blocks of S(4,7,23), with points forming a coclique, p ~ B when p is not in B, and B ~ B' when B,B' meet in 1 point. This yields a graph on 276 vertices that is member of the switching class of a regular two-graph. Now switch one point isolated (and delete it) to get Γ. See also h) below.

Subgraphs

We give the substructures of Γ associated to the 11 maximal subgroups of G not containing McL, sorted according to increasing orbit size.

a) Vertices. There are 275 of these, forming a single orbit. The stabilizer of one (in the full automorphism group) is U4(3):2 with vertex orbit sizes 1+112+162. The graphs induced on the first and second subconstituent are described separately.

b) 22-cocliques. There are 4050 of these, forming a single orbit. The stabilizer of one is M22 with vertex orbit sizes 22+77+176. The graph induced on the 77 is the M22-graph.

c) Pairs of Hoffman-Singleton graphs. Γ has an orbit of splits 50+50+175 where the 50's induce the Hoffman-Singleton graph and the 175 induces the graph on the edges of the Hoffman-Singleton graph, adjacent when they are disjoint and lie in the same pentagon. This latter graph is strongly regular with parameters (v,k,λ,μ) = (175,72,20,36). If one takes the union of the two 50's and switches on this split one obtains the Higman-Sims graph. The stabilizer of a split 50+50+175 is U3(5):2, where the outer 2 interchanges the two 50's. There are 7128 such splits. *** Other HoSi's in Γ? ***

d) 5-cliques. There are 15400 of these, forming a single orbit. The stabilizer of one is 3+1+4:4S5 with vertex orbit sizes 5+270.

e) Edges. There are 15400 of these, forming a single orbit. The stabilizer of one is 34:(M10 × 2) with vertex orbit sizes 2+30+81+162. The graph induced on the 81 is the VO4(3) graph.

f) Non-edges. There are 22275 of these, forming a single orbit. The stabilizer of one is L3(4):22 with vertex orbit sizes 2+56+105+112. The graph induced on the 56 (the common neighbours) is the Gewirtz graph.

g) Odd graphs. Γ contains an orbit of 22275 Odd graphs O4. The stabilizer of one is 2.S8 with vertex orbit sizes 35+240. There are other subgraphs O4 in Γ since the local graph also contains such graphs. *** Any further Odd subgraphs? ***

h) Maximal 7-cocliques. There are 44550 of these, forming a single orbit. The stabilizer of one is 24:A7 with vertex orbit sizes 7+16+112+140. The 16 here is a maximal 16-coclique. (This is what one gets from the regular two-graph construction after switching a block isolated and deleting it.)

i) Maximal 11-cocliques of the first kind. There are 113400 of these, forming a single orbit. Γ has three orbits of maximal 11-cocliques. The first has stabilizer 2 × M11, with vertex orbit sizes 11+22+110+132. The 11 is a maximal 11-coclique, the 22 a (nice) 11K2. (There is a unique nice 11K2 on each 2K2, but there are many others as well.) The other two orbits of maximal 11-cocliques have stabilizers of order 12 and 36, not transtive on the coclique.

j) Splits 125+150. Γ has an orbit of splits 125+150 with stabilizer 5+1+2:3:8:2 with vertex orbit sizes 125+150. The 125 induces a rank 5 strongly regular graph with parameters (v,k,λ,μ) = (125,52,15,26).

k) 3-Cocliques. There are 779625 of these, forming a single orbit. The stabilizer of one is 22+4:(S3×S3) with vertex orbit sizes 3+8+32+64+72+96. The orbit of size 8 induces a K4,4.

Cocliques

The largest cliques have size 5. (It is unknown whether one can select some of the cliques in order to make Γ the collinearity graph of a partial geometry.)

The largest cocliques have size 22. Numbers of maximal cocliques of given size (with orbit sizes):

7: 44550
10: 13721400 = 1247400 + 12474000
11: 199697400 = 113400 + 49896000 + 149688000
13: 43659000 = 6237000 + 37422000
16: 757350 = 44550 + 712800
22: 4050

The largest cocliques in the line graph have size 11 (and there are 254016000 of these, 181440 on each edge, 2240 on each pair of disjoint edges).

References

A.E. Brouwer & J.H. van Lint,
Strongly regular graphs and partial geometries,
pp 85-122 in: Enumeration and Design - Proc. Silver Jubilee Conf. on Combinatorics, Waterloo, 1982, D.M. Jackson & S.A. Vanstone (eds.), Academic Press, Toronto, 1984.

J.-M. Goethals & J. J. Seidel,
The regular two-graph on 276 vertices,
Discr. Math. 12 (1975) 143-158.