On the mean subtree order of graphs under edge addition

For a graphG, themean subtree orderofGis the average order of a subtree ofG. In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graphGand every pair of distinct verticesuandvofG, the addition of the edge betweenuandvincreases 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 orderncan decrease the mean subtree order by as much asn/3asymptotically. We propose the weaker conjecture that for every connected graphGwhich is not complete, there exists a pair of nonadjacent verticesuandv, such that the addition of the edge betweenuandvincreases the mean subtree order. We prove this conjecture in the special case thatGis a tree.