I think it might help to think of the following definition of a Ramanujan graph - a graph whose non-trivial eigenvalues are such that their magnitude is bounded above by the spectral radius of its universal cover.

By "non-trivial eigenvalues" I mean all the eigenvalues except the highest and the smallest. A universal cover of a graph is the infinite tree such that every connected lift of the graph is a quotient of the tree. The spectral radius of a graph would be the norm of its adjacency matrix.

It would be helpful if people can give any pointers along these directions..