More balancing for distance-regular graphs

Let Γ be a distance-regular graph with diameter d ⩾ 2 and let its intersection array be { b 0 , b 1 , … , b d − 1 ; c 1 , … , c d } . For a given eigenvalue θ of Γ and the corresponding minimal idempotent E with the corresponding cosine sequence ω 0 , … , ω d , the following inequality holds c i ( ω 2 + ω i − ( ω 1 + ω i − 1 ) 2 1 + ω i ) + b i − 1 ( ω 2 + ω i − 1 − ( ω 1 + ω i ) 2 1 + ω i − 1 ) ⩾ ( k − θ ) ( ω 1 + ω 2 + ω i − 1 + ω i ) − ( θ + 1 ) ( 1 − ω 2 ) , for any integer i ( 2 ⩽ i ⩽ d ) such that − 1 ∉ { ω i − 1 , ω i } , with equality if and only if for all vertices x , y ∈ V Γ with ∂ ( x , y ) = j + ε , the vectors E ( x + y ) and E ( ∑ z ∈ Γ ( x ) ∩ Γ j − ε ( y ) z + ∑ z ′ ∈ Γ j − ε ( x ) ∩ Γ ( y ) z ′ ) are collinear, where ε = ± 1 2 and j = i − 1 2 . 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.