<正> The h-super connectivity ~kh and the h-super edge-connectivity λh are more refined network relia-bility indices than the connectivity and the edge-connectivity.This paper shows that for a connected balanceddigraph D and its line digraph L,if D is optimally super edge-connected,then k_1(L)=2λ_1(D),and that fora connected graph G and its line graph L,if one of k_1(L)and λ_2(G)exists,then k_1(L)=λ_2(G).This paperdetermines that k_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 thatk_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) areboth super connected for any d≥2 and n≥1.
n阶连通图G的路由选择R是由连接G的每个有向顶点对的n(n-1)条路组成.R经过G的每个顶点(每条边)的路的最大条数称为G关于R的点转发指数ξ(G,R)(边转发指数π(G,R)).对G的所有路由选择R,ξ(G,R)(π(G,R))的最小值称为G的点转发指数ξ(G)(边转发指数π(G)).对于k正则k连通图G,Fernandez de la Vega和Manoussakis[Discrete Applied Mathematics,1989,23(2) :103-123]证明ξ(G)≤(n-1)·┌(n-k-1)/k┐和π(G)≤n┌(n-k-1)/k┐,并且猜想ξ(G)≤┌(n-k)(n-k-1)/k┐.我们分别改进了ξ(G)≤(n-1) ┌(n-k-1)/k┐-(n-k-1)和π(G)≤n┌(n-k-1)/k┐-(n-k),并且证明了猜想对k=3的情形.
