So, using some values off the Web, notably 0.0077g/m^2 for the weight of graphene, and computing the mass of a spherical shell at Pluto's radius, roughly -- 40 AU from a central star, I get (using GNU Units):
You have: 4 * pi * 40au^2 * 0.0077g/m^2
You want: million gigatonne
* 86.618649
/ 0.011544858
How's that compare to the mass of one typical rocky planet?
You have: earthmass / 86.6 million gigatonne
You want:
Definition: 68986.619
So, 1/70,000th an Earth Mass.
Turns out that there's not all that much carbon on Earth -- it's 730 ppm, but that's still a total abundance of about 50x what would be needed for our Pluto-diameter shell.
On the other hand, that means that at best, using an entire Earth's worth of carbon, you could create a Pluto-orbital-radius graphene Dyson sphere 50 atoms thick.
But Pluto's kinda way out there. What if we make the sphere the size of Earth's orbit instead. Surely that's going to save us a few orders of magnitude of matter, right?
Not quite. About 1.5 OOM. We go from ~90 million gigatonne to 2 million gigatonne. The difference between squares and cubes matters. So, on the positive side, you get about 2,000 layers of graphene, but you're still using up all of Earth's available carbon to do that.
There are more abundant minerals -- silicon or magnesium or iron, for example. But you'd still be using planetary-mass quantities for any substantial structure (or multiples of structures orbiting independently and forming a sphere).
Which is what Dyson's also proposed. And my thought is that a Dyson "sphere" comprised of, say, multiple flat sheets of stuff orbiting a star, would tend to have a few issues, such as solar wind, and occasional emissions from incomplete closure. You've also got the question of how you're going to move all that captured energy from where it's been captured to where you plan to use it.
And you still need a cold end to your hot end.
https://en.m.wikipedia.org/wiki/Abundance_of_the_chemical_el...
http://www.geek.com/science/geek-answers-what-is-graphene-15...
http://www.mathopenref.com/spherearea.html
http://hyperphysics.phy-astr.gsu.edu/hbase/solar/soldata3.ht...