As the nail plate grows forward, it should push this portion, along with the graphite, off the free edge. Do not try to clean it out! You might want to put some alcohol or peroxide under there to help prevent infection from developing through the open tear between the nail plate and the nail bed.
It isn't life treatening

I got 'stabbed' with a pencil both in my forehead and under my bottom lip. I still have the 'stain' from the graphite and they look like little blue pindots, but I am healthy as a horse!