The average order of a subtree of a tree

AbstractLet T be a tree all of whose internal vertices have degree at least three. In 1983 Jamison conjectured in JCT B that the average order of a subtree of T is at least half the order of T. In this paper a proof is provided. In addition, it is proved that the average order of a subtree of T is at most three quarters the order of T. Several open questions are stated

Random subtrees of complete graphs

We study the asymptotic behavior of four statistics associated with subtrees of complete graphs: the uniform probability $p_n$ that a random subtree is a spanning tree of $K_n$, the weighted probability $q_n$ (where the probability a subtree is chosen is proportional to the number of edges in the subtree) that a random subtree spans and the two expectations associated with these two probabilities. We find $p_n$ and $q_n$ both approach $e^{-e^{-1}}\approx .692$, while both expectations approach the size of a spanning tree, i.e., a random subtree of $K_n$ has approximately $n-1$ edges

The path minimises the average size of a connected induced subgraph

We prove that among all graphs of order n, the path uniquely minimises the average order of its connected induced subgraphs. This confirms a conjecture of Kroeker, Mol and Oellermann, and generalises a classical result of Jamison for trees, as well as giving a new, shorter proof of the latter. While this paper was being prepared, a different proof was given by Andrew Vince.Comment: 9 pages, 1 figure. Changed title, new figure and minor rewritin