For a connected graph G and α ∈ [ 0 , 1 ] , let D α (G) be the matrix D α (G) = α Tr (G) + (1 − α) D (G) , where D (G) is the distance matrix of G and Tr (G) is the diagonal matrix of its vertex transmissions. Let K m be a complete graph of order m. For n , s fixed, n > s , let G p = K s ∨ (K p ∪ K n − s − p) be the graph obtained from K s and K p ∪ K n − s − p and the edges connecting each vertex of K s with every vertex of K p ∪ K n − s − p. This paper presents some extremal results on the spectral radius of D α (G) that generalize previous results on the spectral radii of the distance matrix and distance signless Laplacian matrix. Among all connected graphs G on n vertices with a vertex/edge connectivity at most s , it is proved that 1. there exists a unique α _ ∈ (3 4 , 3 n − s 4 n − 2 s) such that if α ∈ [ 0 , α _) then the minimal spectral radius of D α (G) is uniquely attained by G = G 1 , 2. there exists a unique α ‾ ∈ (3 4 , 3 n − s 4 n − 2 s) , α ‾ ≥ α _ , such that if α ∈ (α ‾ , 1) then the minimal spectral radius of D α (G) is uniquely attained by G = G ⌊ n − s 2 ⌋ , and 3. if α = 1 then the minimal spectral radius of Tr (G) is n − 1 + ⌈ n − s 2 ⌉ and it is uniquely attained by G = G ⌊ n − s 2 ⌋. Furthermore, in terms of n and s , a tight lower bound l (n , s) of α _ and a tight upper bound u (n , s) of α ‾ are obtained. Finally, for s fixed, it is observed that lim n → ∞ l (n , s) = lim n → ∞ α _ = lim n → ∞ u (n , s) = lim n → ∞ α ‾ = 3 4. [ABSTRACT FROM AUTHOR]