Distance-regular graphs with light tails

Let ΓΓ be a distance-regular graph with valency k≥3k≥3 and diameter d≥2d≥2. It is well known that the Schur product E∘FE∘F of any two minimal idempotents of ΓΓ is a linear combination of minimal idempotents of ΓΓ. 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 QQ-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=FE=F, the rank one minimal idempotent E0E0 is always present in this linear combination and can be the only one only if E=E0E=E0 or E=EdE=Ed and ΓΓ is bipartite. We study the case when E∘E∈span{E0,H}∖span{E0} for some minimal idempotent HH of ΓΓ. We call a minimal idempotent EE with this property a light tail  . Let θθ be an eigenvalue of ΓΓ not equal to ±k±k and with multiplicity mm. We show that m−kk≥−(θ+1)2a1(a1+1)((a1+1)θ+k)2+ka1b1. Let EE be the minimal idempotent corresponding to θθ. The equality case is equivalent to EE being a light tail. Two additional characterizations of the case when EE 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 ΓΓ with a1a1 and the cosine sequence corresponding to EE. 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.