I have a website which allows login via facebook functionality and displays photos from facebook.

While accessing from a mobile browser I would like the website to automatically login(when the click on FB login button, without entering username and password) if the user is already logged in via the native FB application (iOS or andriod). It seems to be that I can do that by building a native iOS or android application and use facebook single sign on feature. Is it possible to do that without having the user install anything on their mobile device?