Suppose $X_1, ..., X_n$ are i.i.d. random variables uniformly distributed in the unit ball in $\mathbb{R}^m$. Suppose, $\Gamma$ is a graph, in which $X_1, … , X_n$ are the vertices and every edge is drawn between a vertex $X_i$ and the vertex, that is the closest to it in the Euclidean metric.
$\Gamma$ is almost surely a forest (all its connected components are trees).
Proof:
Suppose $X_{i_1}, ... , X_{i_n}$ form a cycle with non-zero probability. It is not hard to see, that it almost surely had to be an oriented cycle before we "erased" orientation. That means, that we can assume without the loss of generality $X_{i_{k+1 (\mod n)}}$ is always the closest to $X_{i_k}$. Thus, The sequence $||X_{i_{k+1 (\mod n)}}-X_{i_{k (\mod n)}}||$ is almost surely monotonously decreasing. But it is at the same time periodic with period $n$. And here we get a contradiction.
What is the expectation of the largest diameter of connected component of $\Gamma$?
If an exact value is not possible, then asymptotic will also be accepted as a valid answer.