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

Decreasing the mean subtree order by adding kk edges

The mean subtree order of a given graph $G$, denoted $\mu(G)$, is the average number of vertices in a subtree of $G$. Let $G$ be a connected graph. Chin, Gordon, MacPhee, and Vincent [J. Graph Theory, 89(4): 413-438, 2018] conjectured that if $H$ is a proper spanning supergraph of $G$, then $\mu(H) > \mu(G)$. Cameron and Mol [J. Graph Theory, 96(3): 403-413, 2021] disproved this conjecture by showing that there are infinitely many pairs of graphs $H$ and $G$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)|= |E(G)|+1$ such that $\mu(H) < \mu(G)$. They also conjectured that for every positive integer $k$, there exists a pair of graphs $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that $\mu(H) < \mu(G)$. Furthermore, they proposed that $\mu(K_m+nK_1) < \mu(K_{m, n})$ provided $n\gg m$. 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 $T$ of order $n,$ one can contract any edge and obtain a new tree $T^{*}$ of order $n-1.$ In 1983, Jamison made a conjecture that the mean subtree order, i.e., the average order of all subtrees, decreases at least $\frac{1}{3}$ 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

MAXIMISING THE NUMBER OF CONNECTED INDUCED SUBGRAPHS OF UNICYCLIC GRAPHS

Denote by G(n, d, g, k) the set of all connected graphs of order n, having d > 0 cycles, girth g and k pendent vertices. In this paper, we give a partial characterisation of the structure of all maximal graphs in G(n, d, g, k) for the number of connected induced subgraphs. For the special case d = 1, we find a complete characterisation of all maximal unicyclic graphs. We also derive a precise formula for the maximum number of connected induced subgraphs given: (1) order, girth, and number of pendent vertices; (2) order and girth; (3) order