  We study extensions of direct sums of symmetric operators
$S=\oplus_{n\in\mathbb{N}} S_n$. In general there is no natural boundary
triplet for $S^*$ even if there is one for every $S_n^*$, $n\in\mathbb{N}$. We
consider a subclass of extensions of $S$ which can be described in terms of the
boundary triplets of $S_n^*$ and investigate the self-adjointness, the
semi-boundedness from below and the discreteness of the spectrum. Sufficient
conditions for these properties are obtained from recent results on weighted
discrete Laplacians. The results are applied to Dirac operators on metric
graphs with point interactions at the vertices. In particular, we allow graphs
with arbitrarily small edge length.
