The path minimises the average size of a connected induced subgraph

TL;DR

This paper proves that among all graphs of a given size, the path uniquely minimizes the average size of its connected induced subgraphs, confirming a conjecture and generalizing a classical tree result.

Contribution

It establishes a new proof that the path uniquely minimizes the average size of connected induced subgraphs among all graphs of the same order.

Findings

Paths uniquely minimize the average size of connected induced subgraphs

Confirms a conjecture by Kroeker, Mol, and Oellermann

Provides a shorter proof of a classical result for trees

Abstract

We prove that among all graphs of order n, the path uniquely minimises the average order of its connected induced subgraphs. This confirms a conjecture of Kroeker, Mol and Oellermann, and generalises a classical result of Jamison for trees, as well as giving a new, shorter proof of the latter. While this paper was being prepared, a different proof was given by Andrew Vince.