On Automorphisms of a Distance-Regular Graph with Intersection Array {69, 56, 10; 1, 14, 60}

Abstract

Let Γ be a distance-regular graph of diameter 3 with eigenvalues θ₀ > θ₁ > θ₂ > θ₃. If θ₂ = −1, then the graph Γ₃ is strongly regular and the complementary graph \({\bar \Gamma _3}\) is pseudogeometric for pG_c3(k, b₁/c₂). If Γ₃ does not contain triangles and the number of its vertices v is less than 800, then Γ has intersection array {69, 56, 10; 1, 14, 60}. In this case Γ₃ is a graph with parameters (392, 46, 0, 6) and \({\bar \Gamma _2}\) is a strongly regular graph with parameters (392, 115, 18, 40). Note that the neighborhood of any vertex in a graph with parameters (392, 115, 18, 40) is a strongly regular graph with parameters (115, 18, 1, 3) and its existence is unknown. In this paper, we find possible automorphisms of these strongly regular graphs and automorphisms of a hypothetical distance-regular graph with intersection array {69, 56, 10; 1, 14, 60}. In particular, it is proved that the latter graph is not ar-ctransitive.