On graphs with complete multipartite µ-graphs

Jurisic and Koolen proposed to study 1-homogeneous distance-regular graphs, [email protected] (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 @m-graphs, and that the intersection number @a exists, which means that for a triple (x,y,z) of vertices in @C, such that x and y are adjacent and z is at distance 2 from x and y, the number @a(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"x"n denote the complete multipartite graph with t parts, each of which consists of an n-coclique. We show that if @C is a graph whose @m-graphs are all isomorphic to K"t"x"n and whose intersection number @a exists, then @a=t, as conjectured by Jurisic and Koolen, provided @a>=2. We also prove [email protected]?4, and that equality holds only when @C is the unique distance-regular graph 3.O"7(3).