Decreasing the mean subtree order by adding $k$ edges

TL;DR

This paper confirms conjectures that adding a fixed number of edges to a connected graph can decrease its mean subtree order, countering previous assumptions that such additions always increase it.

Contribution

The paper proves two conjectures showing that increasing edges by a fixed amount can reduce the mean subtree order of a graph.

Findings

Counterexamples to the monotonicity of mean subtree order with added edges.

Proof that for every positive integer k, there exist graphs with H⊃G and |E(H)|=|E(G)|+k where μ(H)<μ(G).

Validation of the conjecture that μ(K_m + nK_1) < μ(K_{m,n}) for large n.

Abstract

The mean subtree order of a given graph $G$, denoted $\mu(G)$, is the average number of vertices in a subtree of $G$. Let $G$ be a connected graph. Chin, Gordon, MacPhee, and Vincent [J. Graph Theory, 89(4): 413-438, 2018] conjectured that if $H$ is a proper spanning supergraph of $G$, then $\mu(H) > \mu(G)$. Cameron and Mol [J. Graph Theory, 96(3): 403-413, 2021] disproved this conjecture by showing that there are infinitely many pairs of graphs $H$ and $G$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)|= |E(G)|+1$ such that $\mu(H) < \mu(G)$. They also conjectured that for every positive integer $k$, there exists a pair of graphs $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that $\mu(H) < \mu(G)$. Furthermore, they proposed that $\mu(K_m+nK_1) < \mu(K_{m, n})$ provided $n\gg m$. In this note, we confirm these two conjectures.