The average size of a connected vertex set of a k-connected graph

Abstract

The topic is the average order A(G) of a connected induced subgraph of a graph G. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1983 Jamison proved that the average order, over all trees of order n, is minimized by the path Pn, the average being A(Pn)=(n+2)/3. In 2018, Kroeker, Mol, and Oellermann conjectured that Pn minimizes the average order over all connected graphs G - a conjecture that was recently proved. In this short note we show that this lower bound can be improved if the connectivity of G is known. If G is k-connected, thenA(G)>n2(1−12k+1).