Abstract

Let Γ denote a distance-regular graph with diameter d2, 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 . We characterize these properties algebraically.

The Terwilliger algebra T=T(x) of Γ with respect to x is the matrix subalgebra generated by A, , where A is the adjacency matrix of Γ and 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 of T generated by . We show that Γ is 1-homogeneous with respect to x if and only if (1id-1) and . We also show that when the intersection number a₁≠0, Γ is pseudo-1-homogeneous with respect to x if and only if (1id). We then characterize these properties according to the structure of the summands in the decomposition of into minimal left ideals.

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 (0id). The endpoint of L is . 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.

Keywords: Terwilliger algebra