When I post is published using the application on facebook, there is a small icon below, next to the like button. I want to add application name or some custom text right next to that icon. Below is a screenshot of twitter app that does what I mean. Posts are published using open graph api. Any ideas?

Those are called actions, and you can add them by sending an array of JSON objects as a parameter to your post call.
Example:
Source: Graph API > Post