28 research outputs found
Random subtrees of complete graphs
We study the asymptotic behavior of four statistics associated with subtrees
of complete graphs: the uniform probability that a random subtree is a
spanning tree of , the weighted probability (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 and both approach ,
while both expectations approach the size of a spanning tree, i.e., a random
subtree of has approximately 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
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
Decreasing the mean subtree order by adding edges
The mean subtree order of a given graph , denoted , is the average
number of vertices in a subtree of . Let be a connected graph. Chin,
Gordon, MacPhee, and Vincent [J. Graph Theory, 89(4): 413-438, 2018]
conjectured that if is a proper spanning supergraph of , then . Cameron and Mol [J. Graph Theory, 96(3): 403-413, 2021] disproved this
conjecture by showing that there are infinitely many pairs of graphs and
with , and such that . They also conjectured that for every positive integer , there
exists a pair of graphs and with , and such that . Furthermore, they proposed that
provided . In this note, we confirm
these two conjectures.Comment: 11 Pages, 5 Figures Paper identical to JGT submissio
On the difference of mean subtree orders under edge contraction
Given a tree of order one can contract any edge and obtain a new
tree of order In 1983, Jamison made a conjecture that the mean
subtree order, i.e., the average order of all subtrees, decreases at least
in contracting an edge of a tree. In 2023, Luo, Xu, Wagner and
Wang proved the case when the edge to be contracted is a pendant edge. In this
article, we prove that the conjecture is true in general