I think this graph has a name: the vertices are bit strings of length $n$, and $(x_1, \ldots , x_n)$ is connected to $(x_2, \ldots, x_n, 0)$, $(x_2, \ldots, x_n, 1)$, $(0,x_1, \ldots , x_{n1})$ and $(1, x_1, \ldots , x_{n1})$. I'm wondering (a) what the name is and (b) where I can read more about this graph. Thanks!

They are called De Bruijn graphs (De Bruijn graphs are generally considered directed, and can be defined over any set of symbols, not just $\{0,1\}$). 

