Average connectivity of minimally 2-connected graphs and average edge-connectivity of minimally 2-edge-connected graphs

Let G be a (multi)graph of order n and let u,v vertices G. The maximum number internally disjoint u–v paths in is denoted by κG(u,v), the edge-disjoint λG(u,v). average connectivity defined κ¯(G)=∑κG(u,v)∕n2, edge-connectivity λ¯(G)=∑λG(u,v)∕n2, where both sums run over all unordered pairs {u,v}⊆V(G). A graph called ideally connected if κG(u,v)=min{deg(u),deg(v)} for {u,v} We prove that every minimally 2-connected with largest bipartite, set degree 2 at least 3 being partite sets. use this structure to κ¯(G)<94 any This bound asymptotically tight, we extremal obtained from some nearly regular on roughly n∕4 3n∕4 edges subdividing edge. also λ¯(G)<94 2-edge-connected G, provide similar characterization graphs.