On graphs with complete multipartite -graphs

Jurišić and Koolen proposed to study 1-homogeneous distance-regular graphs, whose μ -graphs (that is, the graphs induced on the common neighbours of two vertices at distance 2) are complete multipartite. Examples include the Johnson graph J (8, 4), the halved 8-cube, the known generalized quadrangle of order (4, 2), an antipodal distance-regular graph constructed by T. Meixner and the Patterson graph. We investigate a more general situation, namely, requiring the graphs to have complete multipartite μ -graphs, and that the intersection number α exists, which means that for a triple ( x , y , z ) of vertices in Γ , such that x and y are adjacent and z is at distance 2 from x and y , the number α ( x , y , z ) of common neighbours of x , y and z does not depend on the choice of a triple. The latter condition is satisfied by any 1-homogeneous graph. Let K t × n denote the complete multipartite graph with t parts, each of which consists of an n -coclique. We show that if Γ is a graph whose μ -graphs are all isomorphic to K t × n and whose intersection number α exists, then α = t , as conjectured by Jurišić and Koolen, provided α ≥ 2 . We also prove t ≤ 4 , and that equality holds only when Γ is the unique distance-regular graph 3 . O 7 ( 3 ) .