On the Mean Subtree Order of Graphs Under Edge Addition
Ben Cameron, Lucas Mol

TL;DR
This paper investigates how adding edges affects the average size of subtrees in a graph, providing counterexamples to a previous conjecture and proposing a weaker, proven conjecture for trees.
Contribution
It disproves a recent conjecture that edge addition always increases mean subtree order and introduces a weaker conjecture, which is proven for trees.
Findings
Adding an edge can decrease mean subtree order by up to n/3
Counterexamples show the original conjecture is false
The weaker conjecture holds for trees
Abstract
For a graph , the mean subtree order of is the average order of a subtree of . In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph and every pair of distinct vertices and of , the addition of the edge between and increases the mean subtree order. In fact, we show that the addition of a single edge between a pair of nonadjacent vertices in a graph of order can decrease the mean subtree order by as much as asymptotically. We propose the weaker conjecture that for every connected graph which is not complete, there exists a pair of nonadjacent vertices and , such that the addition of the edge between and increases the mean subtree order. We prove this conjecture in the special case that is a tree.
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Taxonomy
TopicsGraph theory and applications · Advanced Graph Theory Research · Limits and Structures in Graph Theory
