6 problems

Let $m$ and $n$ be positive integers. Write $K_m+nK_1$ for the disjoint union of a complete graph on $m$ vertices and $n$ isolated vertices, and let $K_{m,n}$ be the complete bipar…

Cameron–Mol's conjecture. For every positive integer $k$, there exist such graphs satisfying

Let $G$ be a simple connected graph, and let $H$ be a proper spanning supergraph of $G$. The mean subtree order $mu(G)$ is the average order of a subtree of $G$. Chin–Gordon–MacPhe…

Let $G_n$ be a sequence of connected graphs, each with $n$ vertices and $O(n^{\alpha})$ edges, where $\alpha<2$ is fixed. Here $P(G)$ denotes the probability that a uniformly chose…

Let $G_n$ be a sequence of connected graphs, each with $n$ vertices and $Omega(n^2)$ edges, and suppose that each $G_n$ is edge-transitive. Chin et al.'s conjecture. Then … Here…

Conjecture on arbitrary subtrees. The average size of an arbitrary subtree is asymptotically normal, as shown by Wagner in the corresponding uniform random labelled-tree case.