h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Luminosity is related but does not mean radius (size)
In the Main Sequence, the 4 things are related:
Massive stars are big, hotter and brighter;
Low mass stars are small, cooler and less bright.
Color is directly linked to surface temperature;
Luminosity depends on size, but it depends even more on temperature; that's why the Main Sequence exists.
red Giant stars are aged stars, which became larger. This increases luminosity. But the surface is more distant from the energy source inside, so it cools and become red. This reduces luminosity. Since the change in radius is more dramatic than the change in temperature, the stars actually bright more stronger.