  We study the problem of approximating the value of the matching polynomial on
graphs with edge parameter $\gamma$, where $\gamma$ takes arbitrary values in
the complex plane.
  When $\gamma$ is a positive real, Jerrum and Sinclair showed that the problem
admits an FPRAS on general graphs. For general complex values of $\gamma$,
Patel and Regts, building on methods developed by Barvinok, showed that the
problem admits an FPTAS on graphs of maximum degree $\Delta$ as long as
$\gamma$ is not a negative real number less than or equal to
$-1/(4(\Delta-1))$. Our first main result completes the picture for the
approximability of the matching polynomial on bounded degree graphs. We show
that for all $\Delta\geq 3$ and all real $\gamma$ less than $-1/(4(\Delta-1))$,
the problem of approximating the value of the matching polynomial on graphs of
maximum degree $\Delta$ with edge parameter $\gamma$ is #P-hard.
  We then explore whether the maximum degree parameter can be replaced by the
connective constant. Sinclair et al. showed that for positive real $\gamma$ it
is possible to approximate the value of the matching polynomial using a
correlation decay algorithm on graphs with bounded connective constant (and
potentially unbounded maximum degree). We first show that this result does not
extend in general in the complex plane; in particular, the problem is #P-hard
on graphs with bounded connective constant for a dense set of $\gamma$ values
on the negative real axis. Nevertheless, we show that the result does extend
for any complex value $\gamma$ that does not lie on the negative real axis. Our
analysis accounts for complex values of $\gamma$ using geodesic distances in
the complex plane in the metric defined by an appropriate density function.
