(Have only had the mobile version for one day, so if this already built in, sorry)
I don't know if it would be possible (maybe something for the future), to add the social media buttons either on the title or at the bottom of each (or just the home) page. (twitter, facebook...etc. Like on the regular version of Graphene)
I realize I could add them to the menu/footer menu by creating custom links, but maybe have some small versions of the buttons on the main theme.