Solution to a conjecture on the mean subtree order of graphs under edge addition

Abstract

The mean subtree order of a graph is the number of vertices of all subtrees of this graph divided by the number of all subtrees of this graph. In 2021, Cameron and Mol constructed a special graph of order and proved that the addition of a single edge between a pair of nonadjacent vertices in the graph can decrease the mean subtree order by as much as when is large enough. Moreover, they left a conjecture that for every positive integer , there exists a graph obtained by adding edges to another graph , then the mean subtree order of is more than that of . In this paper, we confirm this conjecture.