# On the extrema of the mean subtree order of graphs

## TL;DR

This paper investigates the extremal values of the mean subtree order in connected graphs, confirming that the path minimizes it and exploring approaches to prove the clique maximizes it.

## Contribution

It proves that the path graph attains the minimum mean subtree order among connected graphs, and discusses potential methods to establish the clique as the maximum.

## Key findings

- Path graph minimizes mean subtree order
- Discussion of approaches to prove clique maximizes it
- Extension of previous ideas by Haslegrave and Vince

## Abstract

It has been conjectured that the minimum and maximum of the mean subtree order among connected graphs of order $n$ are attained by the path $P_n$ and clique $K_n$, respectively. Extending ideas due to Haslegrave and Vince, we confirm that the minimum is indeed attained by $P_n$. On the other hand, we discuss different approaches (both promising and flawed) that could lead to a proof of the extremality of $K_n$.