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.
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 VO–4(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.
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).
J.-M. Goethals & J. J. Seidel,
The regular two-graph on 276 vertices,
Discr. Math. 12 (1975) 143-158.
W. H. Haemers & V. D. Tonchev,
Spreads in strongly regular graphs,
Designs, Codes and Cryptography 8 (1996) 145-157.
Patric R. J. Östergård & Leonard H. Soicher,
There is No McLaughlin Geometry, arXiv:1607.03372, Jul 2016.