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 be vertices of G. The maximum number of internally disjoint u-v paths in G is denoted by kappa(G) (u, v), and the maximum number of edge-disjoint u-v paths in G is denoted by lambda G (u, v). The average connectivity of G is defined by (kappa) over bar (G) = Sigma kappa(G)(u, v)/((2) (n)), and the average edge-connectivity of G is defined by (lambda) over bar (G) = Sigma lambda G (u, v)/((n)(2)), where both sums run over all unordered pairs of vertices {u, v} subset of V(G). A graph G is called ideally connected if kappa(G) (u, v) = min{deg(u), deg(v)} for all unordered pairs of vertices {u, v} of G.We prove that every minimally 2-connected graph of order n with largest average connectivity is bipartite, with the set of vertices of degree 2 and the set of vertices of degree at least 3 being the partite sets. We use this structure to prove that (kappa) over bar (G) < 9/4 for any minimally 2-connected graph G. This bound is asymptotically tight, and we prove that every extremal graph of order n is obtained from some ideally connected nearly regular graph on roughly n/4 vertices and 3n/4 edges by subdividing every edge. We also prove that <(lambda)over bar>(G) < 9/4 for any minimally 2-edge-connected graph G, and provide a similar characterization of the extremal graphs. (C) 2020 Elsevier B.V. All rights reserved.