  We provide a streamlined construction of the Friedrichs extension of a
densely-defined self-adjoint and semibounded operator $A$ on a Hilbert space
$\mathcal{H}$, by means of a symmetric pair of operators. A \emph{symmetric
pair} is comprised of densely defined operators $J: \mathcal{H}_1 \to
\mathcal{H}_2$ and $K: \mathcal{H}_2 \to \mathcal{H}_1$ which are compatible in
a certain sense. With the appropriate definitions of $\mathcal{H}_1$ and $J$ in
terms of $A$ and $\mathcal{H}$, we show that $(JJ^\star)^{-1}$ is the
Friedrichs extension of $A$. Furthermore, we use related ideas (including the
notion of unbounded containment) to construct a generalization of the
construction of the Krein extension of $A$ as laid out in a previous paper of
the authors. These results are applied to the study of the graph Laplacian on
infinite networks, in relation to the Hilbert spaces $\ell^2(G)$ and
$\mathcal{H}_{\mathcal E}$ (the energy space).
