It seems you need to require more than just a verified email address before a user can send invites, ideally something that shows the user has participated in your site in some way. Without knowing what your site is it's hard to give specifics, but the StackOverflow equivalent would be requiring users to have at least X reputation before they can invite others. If you're running a forum you could require that they've made at least X posts.
I'd also suggest a small time limit before new accounts can invite - e.g. they have to have been a member for at least X days. This complicates automated invites somewhat.