Facebook has introduced Graph Search, the all new social search engine that makes it so easy for any facebook user to find and connect with people based on interests, location and so on. This all new facebook graph search seems interesting but yet to roll out to the public. You can request facebook graph search invite in facebook website itself.
Scroll to the bottom of the page in the above link and click on “Join Waiting List“. Facebook will rollout limited invites to users who have joined waiting list in coming days. It may take a day or weeks to get the invite and we have no confirmation about the approximate time that facebook will rollout this new graph search feature to the public.
So, make sure you have submitted your request to join waiting list and added your account to the queue to get a facebook graph search invite. There is no other way so far to get an invite.