Random subtrees and unimodal sequences in graphs

For a complete graph Kn and a nonnegative integer k, we study the probability that a random subtree of Kn has exactly n - k vertices and show that it approaches a limiting value of e-k-e-1 k! as n tends to infinity. We also consider the (conditional) probability that a random subtree of Kn contains a given edge, and more generally, a fixed subtree. In particular, if e and f are adjacent edges of Kn, Chin, Gordon, MacPhee and Vincent [J. Graph Theory 89 (2018), 413-438] conjectured that P[e & SUBE; T | f & SUBE; T] & LE; P[e & SUBE; T]. We prove this conjecture and further prove that P[e & SUBE; T | f & SUBE; T] tends to three-quarters of P[e & SUBE; T] as n & RARR; & INFIN;. Finally, several classes of graphs are given, such as star plus an edge, lollipop graph and glasses graph, whose subtree polynomials are unimodal.& COPY; 2023 Elsevier B.V. All rights reserved.