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 !
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 ! 


Lubotzky and Zuk's book on property ($\tau$) 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 Discrete groups, expanding graphs and invariant measures, 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. 


László Lovász's 2009 book Geometric representations of graphs has a section on Circle packing and the Riemann Mapping Theorem (p.89). I am not certain if it sufficiently connects "to spectral problems in hyperbolic Riemann surfaces" for your purposes, but as it is available as PDF, you could easily check. A great book regardless! 


There's also this one: 

