Distance-regular graphs with light tails

Let Γ be a distance-regular graph with valency k ≥ 3 and diameter d ≥ 2 . It is well known that the Schur product E ∘ F of any two minimal idempotents of Γ is a linear combination of minimal idempotents of Γ . Situations where there is a small number of minimal idempotents in the above linear combination can be very interesting, since they usually imply strong structural properties, see for example Q -polynomial graphs, tight graphs in the sense of Jurišić, Koolen and Terwilliger, and 1- or 2-homogeneous graphs in the sense of Nomura. In the case when E = F , the rank one minimal idempotent E 0 is always present in this linear combination and can be the only one only if E = E 0 or E = E d and Γ is bipartite. We study the case when E ∘ E ∈ span { E 0 , H } ∖ span { E 0 } for some minimal idempotent H of Γ . We call a minimal idempotent E with this property a light tail. Let θ be an eigenvalue of Γ not equal to ± k and with multiplicity m . We show that m − k k ≥ − ( θ + 1 ) 2 a 1 ( a 1 + 1 ) ( ( a 1 + 1 ) θ + k ) 2 + k a 1 b 1 . Let E be the minimal idempotent corresponding to θ . The equality case is equivalent to E being a light tail. Two additional characterizations of the case when E is a light tail are given. One involves a connection between two cosine sequences and the other one a parameterization of the intersection numbers of Γ with a 1 and the cosine sequence corresponding to E . We also study distance partitions of vertices with respect to two vertices and show that the distance-regular graphs with light tails are very close to being 1-homogeneous. In particular, their local graphs are strongly regular.