  We investigate how a protoplanetary disc's susceptibility to gravitational
instabilities and fragmentation depends on the mass of its host star. We use 1D
disc models in conjunction with 3D SPH simulations to determine the critical
disc-to-star mass ratios at which discs become unstable against fragmentation,
finding that discs become increasingly prone to the effects of self-gravity as
we increase the host star mass. The actual limit for stability is sensitive to
the disc temperature, so if the disc is optically thin stellar irradiation can
dramatically stabilise discs against gravitational instability. However, even
when this is the case we find that discs around $2$M$_{\odot}$ stars are prone
to fragmentation, which will act to produce wide-orbit giant planets and brown
dwarfs. The consequences of this work are two-fold: that low mass stars could
in principle support high disc-to-star mass ratios, and that higher mass stars
have discs that are more prone to fragmentation, which is qualitatively
consistent with observations that favour high-mass wide-orbit planets around
higher mass stars. We also find that the initial masses of these planets
depends on the temperature in the disc at large radii, which itself depends on
the level of stellar irradiation.
