On the automorphism group of the Aschbacher graph

On automorphisms of strongly regular graphs with $\lambda=0$ and $\mu=3$

“…On the other hand, a coclique orbit outside the neighborhood of a fixed vertex is connected by thick edges with at most 9 orbits of type (1), and a coclique orbit in the neighborhood of a fixed vertex is connected by thick edges with at most 6 orbits of type (1); this contradicts the fact that the number of thick edges between orbits of type (1) and coclique orbits does not exceed 3 · 6 + 2 · 9 = 36. Suppose there are δ coclique orbits that contain vertices adjacent to triples of vertices in orbits of type (1). None of these δ orbits can contain a vertex adjacent to triples of vertices in different orbits of type (1).…”

“…Let X i be the set of vertices in Γ − Ω that are adjacent to precisely i vertices in Ω, x i = |X i |. (1) x 0 = α 1 (t), 3|Ω| − x 0 is divisible by 9, and if u ∈ X 3 and {b 1 Proof. Let p = 2.…”

“…Suppose there are δ coclique orbits that contain vertices adjacent to triples of vertices in orbits of type (1). None of these δ orbits can contain a vertex adjacent to triples of vertices in different orbits of type (1). Therefore, the number of 2-thick edges between orbits of type (1) and coclique orbits is at least (12 − δ)(2 + 3) − 15 = 45 − 5δ and at most (3 · 6 + 2 · 9) − 3δ = 36 − 3δ.…”

“…For example, the subgraph of fixed points of an automorphism of a Moore graph is itself a Moore graph or a star (see [1,Lemma 1]). The automorphisms of strongly regular graphs with λ = 0 and μ = 2 were considered in [2].…”