More balancing for distance-regular graphs

Let ΓΓ be a distance-regular graph with diameter d⩾2d⩾2 and let its intersection array be {b0,b1,…,bd−1;c1,…,cd}{b0,b1,…,bd−1;c1,…,cd}. For a given eigenvalue θθ of ΓΓ and the corresponding minimal idempotent EE with the corresponding cosine sequence ω0,…,ωdω0,…,ωd, the following inequality holds ci(ω2+ωi−(ω1+ωi−1)21+ωi)+bi−1(ω2+ωi−1−(ω1+ωi)21+ωi−1)⩾(k−θ)(ω1+ω2+ωi−1+ωi)−(θ+1)(1−ω2), for any integer ii(2⩽i⩽d)(2⩽i⩽d) such that −1∉{ωi−1,ωi}−1∉{ωi−1,ωi}, with equality if and only if for all vertices x,y∈VΓx,y∈VΓ with ∂(x,y)=j+ε∂(x,y)=j+ε, the vectors E(x+y)andE(∑z∈Γ(x)∩Γj−ε(y)z+∑z′∈Γj−ε(x)∩Γ(y)z′) are collinear, where ε=±12 and j=i−12. The cases where equality holds are analyzed and new conditions for the vanishing of certain Krein parameters for strongly regular graphs are obtained. In addition, new results for strongly balanced graphs are also presented.