Hello,


Could you name a couple of books or downloadable lecture notes that discuss spectral graph theory and its connection to spectral problems in hyperbolic Riemann surfaces ? You could also mention some papers if you know.


Thank you !



Books that discuss spectral graph theory and its connection to eigenvalue problems in hyperbolic geometry
Lubotzky and Zuk's book [on property ($\tau$)](http://www.ma.huji.ac.il/~alexlub/) discusses expander graphs and the minimal eigenvalue of the Laplacian on covers of Riemann surfaces. See for example Prop. 2.9 in the book. There's also his book D[iscrete groups, expanding graphs and invariant measures](http://books.google.com/books?id=aNURlzNuotEC&lpg=PA1&dq=lubotzky%2520expanders&pg=PP1#v=onepage&q=lubotzky%2520expanders&f=false), but it's not available online. I don't know of relations between eigenvalues of graphs and eigenvalues of surfaces deeper into the spectrum, since the correspondence is only coarse. The papers of Robert Brooks and his collaborators are quite readable. 


