  Main sequence stars exhibit a clear rotation-activity relationship, in which
rapidly rotating stars drive strong chromospheric/coronal ultraviolet and X-ray
emission. While the vast majority of red giant stars are inactive, a few
percent exhibit strong ultraviolet emission. Here we use a sample of 133 red
giant stars observed by SDSS APOGEE and GALEX to demonstrate an empirical
relationship between NUV excess and rotational velocity (vsini). Beyond this
simple relationship, we find that NUV excess also correlates with rotation
period and with Rossby number in a manner that shares broadly similar trends to
those found in M dwarfs, including activity saturation among rapid rotators.
Our data also suggest that the most extremely rapidly rotating giants may
exhibit so-called "super-saturation", which could be caused by centrifugal
stripping of these stars rotating at a high fraction of breakup speed. As an
example application of our empirical rotation-activity relation, we demonstrate
that the NUV emission observed from a recently reported system comprising a red
giant with a black hole companion is fully consistent with arising from the
rapidly rotating red giant in that system. Most fundamentally, our findings
suggest a common origin of chromospheric activity in rotation and convection
for cool stars from main sequence to red giant stages of evolution.
