  There has recently been much interest in Gaussian fields on linear networks
and, more generally, on compact metric graphs. One proposed strategy for
defining such fields on a metric graph $\Gamma$ is through a covariance
function that is isotropic in a metric on the graph. Another is through a
fractional-order differential equation $L^{\alpha/2} (\tau u) = \mathcal{W}$ on
$\Gamma$, where $L = \kappa^2 - \nabla(a\nabla)$ for (sufficiently nice)
functions $\kappa, a$, and $\mathcal{W}$ is Gaussian white noise. We study
Markov properties of these two types of fields. First, we show that no Gaussian
random fields exist on general metric graphs that are both isotropic and
Markov. Then, we show that the second type of fields, the generalized
Whittle--Mat\'ern fields, are Markov if and only if $\alpha\in\mathbb{N}$.
Further, if $\alpha\in\mathbb{N}$, a generalized Whittle--Mat\'ern field $u$ is
Markov of order $\alpha$, which means that the field $u$ in one region
$S\subset\Gamma$ is conditionally independent of $u$ in $\Gamma\setminus S$
given the values of $u$ and its $\alpha-1$ derivatives on $\partial S$.
Finally, we provide two results as consequences of the theory developed: first
we prove that the Markov property implies an explicit characterization of $u$
on a fixed edge $e$, revealing that the conditional distribution of $u$ on $e$
given the values at the two vertices connected to $e$ is independent of the
geometry of $\Gamma$; second, we show that the solution to $L^{1/2}(\tau u) =
\mathcal{W}$ on $\Gamma$ can obtained by conditioning independent generalized
Whittle--Mat\'ern processes on the edges, with $\alpha=1$ and Neumann boundary
conditions, on being continuous at the vertices.
