Conditions for the existence of a graph with given diameter, connectivity, and ball diversity vector

Given some arbitrary integers d ≥ 2, ? ? 1 and an integer vector $ \bar \tau Given some arbitrary integers d ≥ 2, ϰ ⩾ 1 and an integer vector = (τ ₀, τ ₁, …, τ _d ) with τ ₀ ⩾ τ ₁ ⩾ … ⩾ τ _d = 1 and τ _{d − 1} ⩾ d ²ϰ + 3, the existence is proved of a graph of diameter d and connectivity ϰ whose ball diversity vector is . Moreover, the nonexistence is proved of a graph of diameter d with connectivity ϰ and ball diversity vector (τ ₀, τ ₁, …, τ _d ), where τ ₀ < (d − 1)ϰ + 2. Original Russian Text ? K.L. Rychkov, 2007, published in Diskretnyi Analiz i Issledovanie Operatsii, Ser. 1, 2007, Vol. 14, No. 4, pp. 43–56.