A Lower Bound on the Average Size of a Connected Vertex Set of a Graph

TL;DR

This paper proves that among all connected graphs, the path minimizes the average size of a connected induced subgraph, confirming a longstanding conjecture and extending known results from trees to general graphs.

Contribution

It establishes that the path graph uniquely minimizes the average order of connected induced subgraphs among all connected graphs, confirming a conjecture from 2018.

Findings

Paths have the smallest average connected induced subgraph size among all connected graphs.

The conjecture by Kroeker, Mol, and Oellermann is proven true.

The result generalizes previous findings from trees to all connected graphs.

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 1984, Jamison proved that the average order, over all trees of order $n$, is minimized by the path $P_n$. In 2018, Kroeker, Mol, and Oellermann conjectured that $P_n$ minimizes the average order over all connected graphs. The main result of this paper confirms this conjecture.