Do less bright stars have redder spectrum?
Not necessarily. Stars radiate pretty much like black bodies, so the spectrum depends on temperature like so:
The hotter the star, the more it radiates, and the more is the spectrum shifted towards higher frequencies. Colder stars would then normally radiate less and be redder.
But that doesn't take into account the radiative area of the star(so, size). Red giants, for example, are relatively cold, and radiate small amounts of energy per unit surface area, but since the area is so large, they can be extremely bright.
However, stars on the main sequence(http://en.wikipedia.org/wiki/Main_sequence) have their temperatures and sizes dependent on mass only(and age and metallicity, but to a lesser degree). So as long as you restrict yourself to these only, your statement is true.
Thank you very much for such an excellent answer with a diagram and the link to the wikipedia article on Main sequence stars. That was very helpful.
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