Super Connectivity of Line Graphs and Digraphs

The h-super connectivity κh and the h-super edge-connectivity λh 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 κ1(L) = 2λ1(D), and that for a connected graph G and its line graph L, if one of κ1(L) and λ2(G) exists, then κ1(L) = λ2(G). This paper determines that κ1(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 κ1(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.