The h-super connectivity κ h and the h-super edge-connectivity λ h are more refined network reliability indices than the connectivity and the edge-connectivity. This paper shows that for a connected balanced digraph D and its line digraph L, if D is optimally super edge-connected, then κ 1 (L) = 2λ 1 (D), and that for a connected graph G and its line graph L, if one of κ 1 (L) and λ 2 (G) exists, then κ 1 (L) = λ 2 (G). This paper determines that κ 1 (B(d, n)) is equal to 4d − 8 for n = 2 and d ≥ 4, and to 4d − 4 for n ≥ 3 and d ≥ 3, and that κ 1 (K(d, n)) is equal to 4d − 4 for d ≥ 2 and n ≥ 2 except K(2, 2). It then follows that B(d, n) and K(d, n) are both super connected for any d ≥ 2 and n ≥ 1.