Graphene space elevator possible? I just read this story on MIT working on industrial scale, km^2 sheet production of graphene.
A quick check of Wikipedia on graphene and Wikipedia on space elevator tells me

Measurements have shown that graphene has a breaking strength 200 times greater than steel, with a tensile strength of 130 GPa (19,000,000 psi)

and

The largest holdup to Edwards' proposed design is the technological limit of the tether material. His calculations call for a fiber composed of epoxy-bonded carbon nanotubes with a minimal tensile strength of 130 GPa (19 million psi) (including a safety factor of 2)

Does this mean we may soon actually have the material for a space elevator?
 A: A decent terrestrial space elevator could be built with a material with a tensile strength of 50 Gigapascals (including a decent safety factor), so this material may suffice.
Note that there is no prospect of having one 100,000 km nanotube - they would actually be much shorter (maybe 10 cm) and held together by the much weaker inter-tube molecular bonds (if the strings are long enough, they will bond together billions of times where they touch; if there are enough such contact points, the inter-tube bond can be as strong as you want.
Graphene uses the same carbon-carbon bond as the nanotubes for strength, so it would not surprise me if graphene could be used to create strong tethers. I think that what is really holding the terrestrial space elevator back is the lack of money for elevator-focused R&D on string materials. There is really no other market for these materials, and other uses (such as bullet-proof vests) are not close enough to 
A: The real economics will come into play via electricity. Space based solar transmitting electricity down graphene cables solves our energy crisis basically forever. Once you build the first cable, building more is an order of magnitude cheaper. Once you make that initial investment, the solar farms become trivial, although it will take years if not decades to get them up and running. Inexhaustible, utterly green energy that can be scaled virtually limitlessly- thats the gamechanger for the human race.
A: Most proposed designs of the space elevator are such that the whole structure is under tensile stress from the ground anchor point. In these designs, there are stress limits that constraint the material properties of the ribbon. The calculations (based on geosynchronous height of earth) point to that 130 GPa figure.
There is potentially another design approach in which there is no stress limit required in any point in the structure. In this case, the tensile stress is entirely from the geo synchronous orbit holding up the structure against its weight (rather than the earth holding it up against centrifugal force). You only need to make sure the whole structure is at equilibrium, so the center of mass stay roughly at GEO.  So, you start at GEO, and start each level one at a time. after finishing each level, you adjust your center of mass to stay at equilibrium. Then you proceed to build the next level below the previous one, until you reach ground.
In order to the upper levels to be able to hold the weight of the lower ones, the structure will follow a exponential pattern of joints. If whole elevator structure will have $N$ levels, the ground level (Level 0) will have one link. the next level (Level 1) will have $k$ links, which all sustain the weight from level 0 link. Level 2 links will have $k^2$ links, which sustain each of the $k$ links of the level 1 links. The last level will have $k^N$ links.
So at GEO, the stress is the whole weight of the structure by the cross section area of all links. the area grows as $k^N$ while the weight of the whole structure grows as $\frac{1-k^{N+1}}{1-k}$. So asymptotically the stress as GEO stays under parameter control.
The benefit of this approach is that you even can make the whole structure with normal materials (no stress limit required). Of course the tensile strength of the material chosen still affect the number of links and levels required to make the structure sustain its own weight.
A: @lurscher of course I understand it's from GEO, the fact that GEO is the net zero apparent acceleration point is the reason it would be "unfurled" from GEO.  If your point behind the stages is that it could be carried up in segments, then yes, no one ever argued otherwise.  The only thing your $k^N$ mathematics shows is that it could theoretically be made with any material, regardless of its specific strength.  This is true for any compression structure as well.  There is still a practical problem if the approach results in needing trillions of tons of material. 
A: Though we may be a while away from the ability to make the all important material, we could in fact start working towards this now.  A while ago I penned a small idea for building 2 elevators, the first of which would be a "Lunar Elevator".  With greatly reduced gravity, the tensile strength of the cable would be reduced to well within the art of the possible as there are materials around now which could happily (and cheaply) do the job.  M5 or even normal Kevlar or Spectra promise to be more than strong enough.
Once established then we have our system of moving large amounts of Mass from the Lunar surface to the GEO orbit to create our counterbalance.  This in itself would take time (decades) but allow the team on earth to have developed the graphene tech to allow the second elevator to take place.  
I have some workings on the idea if you're keen, though I'm aware I'm not the first.
A: You can't just state the conditions for the material to build a space elevator in terms of tensile strength.  The important quantity is the ratio of tensile strength to mass density.  In the typical design, where the cable is thickest at geosynchronous orbit altitude, you can predict the needed thickness of the cable as a function of this ratio.  It turns out to be an exponential function of the tensile strength/density ratio, and so relatively small differences in the ratio correspond to huge differences in required strengths.  For example, if you do the calculation for steel you find that it is "possible" to build a space elevator that is 1 cm thick at the base as long as it is about 3000 light years (!!!) thick at geosynchronous altitude.  So, a rather interesting meaning of the word "possible".  By using a material with a higher ratio of tensile strength/density you can bring down the ratio of thickness at the ground to thickness at geosynchronous.  Some examples of this ratio (hard calculation, and you'll see different figures if you look in different places):
steel: $3.1 \times 10^{21}$  (1 cm at ground translates to 3000 light years at thickest point - totally impossible)
kevlar: 16000  (1 cm at ground translates to 160 m at geosynchronous - maybe possible but certainly not practical...)
graphene: 6 (1 cm at ground translates to 6 cm at geosynchronous - but only if theories of tensile strength of graphene in bulk are correct...)
If you want to see how to calculate these check out the classic paper by Pearson: https://www.researchgate.net/profile/Jerome-Pearson/publication/222451319_The_orbital_tower_A_spacecraft_launcher_using_the_Earth%27s_rotational_energy/links/5c7d6a31a6fdcc4715af6616/The-orbital-tower-A-spacecraft-launcher-using-the-Earths-rotational-energy.pdf
Note that we don't actually know the tensile strength of graphene very well because we can't yet reliably grow large enough, high quality enough sheets of it to measure this well.  But, even if it is close to the theoretical values then it looks good, if we can synthesize large enough sheets of it.  Some people working on this seem to think woven carbon nanotube fabrics might turn out to be better.  All of this is a long way from being doable because we just can't synthesize these materials in large enough quantities.
