Graphs from the point of view of Riemann surfaces I was listening to the lecture "Graphs from the point of view of
Riemann surfaces" by Prof. Alexander Mednykh. I am looking for references for the basics of this topic. Any kind of suggestions is highly appreciated. Thank you in advance.
 A: I believe one of the first paper was:
Bacher, Roland; de la Harpe, Pierre; Nagnibeda, Tatiana
The lattice of integral flows and the lattice of integral cuts on a finite graph,
Bull. Soc. Math. Fr. 125, No. 2, 167-198 (1997). Zbl 0891.05062
(Sorry, self-promotion).
It has been widely cited by subsequent papers on the subject
(use MathSciNet or Zentralblatt)
I remember also a one or two interesting papers by Mathew Baker (you can find them by looking through the papers citing the above paper) on the subject.
A: The book Graphs on Surfaces and Their Applications by Sergei K. Lando and Alexander K. Zvonkin
From Amazon page:

Graphs drawn on two-dimensional surfaces have always attracted researchers by their beauty and by the variety of difficult questions to which they give rise. The theory of such embedded graphs, which long seemed rather isolated, has witnessed the appearance of entirely unexpected new applications in recent decades, ranging from Galois theory to quantum gravity models, and has become a kind of a focus of a vast field of research. The book provides an accessible introduction to this new domain, including such topics as coverings of Riemann surfaces, the Galois group action on embedded graphs (Grothendieck's theory of "dessins d'enfants"), the matrix integral method, moduli spaces of curves, the topology of meromorphic functions, and combinatorial aspects of Vassiliev's knot invariants and, in an appendix by Don Zagier, the use of finite group representation theory. The presentation is concrete throughout, with numerous figures, examples (including computer calculations) and exercises, and should appeal to both graduate students and researchers.

