The Average Size of a Connected Vertex Set of a $k$-connected Graph

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 1984, Jamison proved that the average order, over all trees of order $n$, is minimized by the path $P_n$, the average being $A(P_n)=(n+2)/3$. In 2018, Kroeker, Mol, and Oellermann conjectured that $P_n$ 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, then $$A(G) \geq \frac{n}2 \Bigg (1- \frac{1}{2^k+1} \Bigg ).$$