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arXiv · 2508.20593

On the extrema of the mean subtree order of graphs

Stijn Cambie·Jorik Jooken·Stephan Wagner
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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 $P_n$. We also show that the maximum is attained by $K_n$ by proving that the ratio between spanning and almost spanning trees is maximised by the clique and applying a double-counting argument.

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Stijn Cambie, Jorik Jooken, Stephan Wagner. 2026-09-16. On the extrema of the mean subtree order of graphs. https://arxiv.org/abs/2508.20593

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