On automorphisms of distance-regular graphs with intersection arrays {2 r + 1, 2 r − 2, 1; 1, 2, 2 r + 1}

Let Γ be an antipodal graph with intersection array {2r+1, 2r−2, 1; 1, 2, 2r+1}, where 2r(r + 1) ≤ 4096. If 2r + 1 is a prime power, then Mathon’s scheme provides the existence of an arc-transitive graph with this intersection array. Note that 2r + 1 is not a prime power only for r ∈ {7, 17, 19, 22, 25, 27, 31, 32, 37, 38, 42, 43}. We study automorphisms of hypothetical distance-regular graphs with the specified values of r. The cases r ∈ {7, 17, 19} were considered earlier. We prove that, if Γ is a vertex-symmetric graph with intersection array {2r + 1, 2r − 2, 1; 1, 2, 2r +1}, 2r + 1 is not a prime power, and r ≤ 43, then r = 25, 27, or 31.