Automorphisms of a Distance-Regular Graph with Intersection Array {30,22,9;1,3,20}

Abstract

A distance-regular graph Γ of diameter 3 is called a Shilla graph if it has the second eigenvalue θ1 = a3. In this case a = a3 divides k and we set b = b(Γ) = k/a. Koolen and Park obtained the list of intersection arrays for Shilla graphs with b = 3. There exist graphs with intersection arrays {12, 10, 5; 1, 1, 8} and {12, 10, 3; 1, 3, 8}. The nonexistence of graphs with intersection arrays {12, 10, 2; 1, 2, 8}, {27, 20, 10; 1, 2, 18}, {42, 30, 12; 1, 6, 28}, and {105, 72, 24; 1, 12, 70} was proved earlier. In this paper we study the automorphisms of a distance-regular graph Γ with intersection array {30, 22, 9; 1, 3, 20}, which is a Shilla graph with b = 3. Assume that a is a vertex of Γ, G = Aut(Γ) is a nonsolvable group, G¯ = G/S(G), and T¯ is the socle of G¯. Then T¯ ∼= L2(7), A7, A8, or U3(5). If Γ is arc-transitive, then T is an extension of an irreducible F2U3(5)-module V by U3(5) and the dimension of V over F3 is 20, 28, 56, 104, or 288. © 2020 Sverre Raffnsoe. All rights reserved.