The Average Connectivity of Minimally Strong Blocks

Let D=(V(D),A(D)) be a digraph of order n. The digraph D is called a strong block if D is strongly connected and its underlying graph has no cut-vertex. D is called a minimally strong block, if D is a strong block, but D−a is not a strong block for every arc a of A(D). For u,v ∈ V(D), the local connectivity ${\kappa }_D$(u,v) from u to v is the maximum number of internally disjoint directed (u,v)-paths in D. The average connectivity of D is $\overline{\kappa }\left(D\right)=\frac{1}{n\left(n-1\right)}\left[{\displaystyle \sum _{\left(u,v\right)\in V\left(D\right)\times V\left(D\right)}\kappa D\left(u,v\right)}\right]$. By using the method of degree sequence and ear decomposition, this paper determine some upper bounds of average connectivity among minimally strong blocks in terms of their orders, and conjecture that it is strictly less than 3/2.