1-homogeneous, pseudo-1-homogeneous, and 1-thin distance-regular graphs

Finally, we use these decompositions to describe a related family of distance-regular graphs. Let L denote a minimal left ideal of T. Then L is said to be thin if dimEi*L⩽1(0⩽i⩽d). The endpoint of L is min{i|Ei*L≠0}. The graph Γ is said to be 1-thin with respect to x when every minimal left ideal of T with endpoint 1 is thin. It is known that Γ is 1-thin with respect to x with a unique minimal left ideal of endpoint 1 if and only if Γ is bipartite or almost bipartite (in either case Γ is 1-homogeneous with respect to x). We show that Γ is 1-thin with respect to x with exactly two minimal left ideals of endpoint 1 if and only if Γ is pseudo-1-homogeneous with respect to x and the intersection number a1 is nonzero.