  Random geometric graphs (RGGs) are commonly used to model networked systems
that depend on the underlying spatial embedding. We concern ourselves with the
probability distribution of an RGG, which is crucial for studying its random
topology, properties (e.g., connectedness), or Shannon entropy as a measure of
the graph's topological uncertainty (or information content). Moreover, the
distribution is also relevant for determining average network performance or
designing protocols. However, a major impediment in deducing the graph
distribution is that it requires the joint probability distribution of the
$n(n-1)/2$ distances between $n$ nodes randomly distributed in a bounded
domain. As no such result exists in the literature, we make progress by
obtaining the joint distribution of the distances between three nodes confined
in a disk in $\mathbb{R}^2$. This enables the calculation of the probability
distribution and entropy of a three-node graph. For arbitrary $n$, we derive a
series of upper bounds on the graph entropy; in particular, the bound involving
the entropy of a three-node graph is tighter than the existing bound which
assumes distances are independent. Finally, we provide numerical results on
graph connectedness and the tightness of the derived entropy bounds.
