Sufficient conditions for the existence of a graph with a given variety of balls

It is proved that for each positive integer d and each collection of integers $bar tau $ = (τ ₀, τ ₁ …, τ _d ) such that τ ₀ ? τ ₁ ? … ? τ _d = 1 and τ _d?1 ? d ² + 1, there exists a graph of diameter d whose variety vector of the balls is equal to $bar tau $ ; if d ? 3 then there is no graph of diameter d whose variety vector of balls (τ ₀, τ ₁, …, τ _d ) satisfies the condition τ ₀ = τ ₁ = … = τ _d?1 ? 2d ? 1.