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

Let Γ denote a distance-regular graph with diameter d ⩾ 2 , and fix a vertex x of Γ . Γ is said to be 1-homogeneous (resp. pseudo-1-homogeneous) with respect to x whenever for all integers h and i between 0 and d, inclusive (resp. for all integers h between 0 and d - 1 and i between 0 and d, inclusive) and for all vertices y and z of Γ with ∂ ( x , y ) = h , ∂ ( y , z ) = i , ∂ ( z , x ) = 1 , the number of vertices w of Γ with ∂ ( x , w ) = j , ∂ ( y , w ) = 1 , ∂ ( z , w ) = k is independent of y and z for all j, k ( 0 ⩽ j , k ⩽ d ) . We characterize these properties algebraically. rwilliger algebra T = T ( x ) of Γ with respect to x is the matrix subalgebra generated by A, E 0 * , E 1 * , … , E d * , where A is the adjacency matrix of Γ and E i * is the diagonal matrix whose nonzero entries are ones in the ( y , y ) positions for those vertices y such that ∂ ( x , y ) = i . Our results concern the left ideal T E 1 * of T generated by E 1 * . We show that Γ is 1-homogeneous with respect to x if and only if dim E i * T E 1 * ⩽ 3 ( 1 ⩽ i ⩽ d - 1 ) and dim E d * T E 1 * ⩽ 2 . We also show that when the intersection number a 1 ≠ 0 , Γ is pseudo-1-homogeneous with respect to x if and only if dim E i * T E 1 * ⩽ 3 ( 1 ⩽ i ⩽ d ) . We then characterize these properties according to the structure of the summands in the decomposition of T E 1 * into minimal left ideals. y, 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 dim E i * L ⩽ 1 ( 0 ⩽ i ⩽ d ) . The endpoint of L is min { i | E i * 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 a 1 is nonzero.