A lower bound on the average size of a connected vertex set of a graph

Abstract

The topic is the average order of a connected induced subgraph of a graph. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1983, Jamison proved that the average order of a subtree, over all trees of order n, is minimized by the path Pn. In 2018, Kroeker, Mol, and Oellermann conjectured that Pn minimizes the average order of a connected induced subgraph over all connected graphs. The main result of this paper confirms this conjecture.