In the original LPS article they mention two constructions for Ramanujan graphs, one with order $q^3$ nodes and the other in order $q$.
My question is regarding the second construction (which is described here at page 5, they call it LPS-II).
What does it mean when they say "in a linear fractional way"? I understand that for a generator $\alpha \in S$, where $S$ is the set of $\ p+1$ generators, and for some integer $z$ - $\alpha(z)$ behaves like so, but what does it mean? how do I actually connect between two vertices?
Hope I am clear enough, if not I will elaborate some more.
Thanks!
They mean that they are viewing the group as a group of permutations of the points on the projective line.
So they are constructing a Cayley graph for the linear fractional group. (This is not a group of matrices, it is the quotient of a matrix group by its centre.)