Distance-regular graphs with intersection array

Abstract

Distance regular graphs Γ of diameter 3 for which the graphs Γ2 and Γ3 are strongly regular, studied by M.S. Nirova. For Q-polynomial graphs with intersection arrays (69; 56; 10; 1; 14; 60) and (119; 100; 15; 1; 20; 105} the graph Γ3 is strongly regular and does not contain triangles. Automorphisms of graphs with these intersection arrays were found by A.A. Makhnev, M.S. Nirova and M.M. Isakova, A.A. Makhnev, respectively. The graph Γ with the intersection array (74; 54; 15; 1; 9; 60) also is Q-polynomial, and Γ3 is a strongly regular graph with parameters (630; 111; 12; 21). It is proved in the paper that graphs with intersection arrays (69; 56; 10; 1; 14; 60), (74; 54; 15; 1; 9; 60) and (119; 100; 15; 1; 20; 105) do not exist. © 2019 Sobolev Institute of Mathematics.