TITLE: Graphs from the point of view of Riemann surfaces
QUESTION [6 upvotes]: 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.

REPLY [10 votes]: It seems that the question is about the following lecture notes:
http://math.nsc.ru/conference/g2/g2r2/files/pdf/Lecture-8.pdf
At the start of these notes, five papers are mentioned. Based on the topic, the authors and the listed years, I believe that these refer to the following papers:

Roland Bacher, Pierre de la Harpe, and Tatiana Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph. Bulletin de la Société Mathématique de France, 125(2):167–198, 1997. https://doi.org/10.24033/bsmf.2303
Hajime Urakawa, A discrete analogue of the harmonic morphism and Green kernel comparison theorems. Glasgow Mathematical Journal, 42(3):319–334, 2000. https://doi.org/10.1017/S0017089500030019
Matthew Baker and Serguei Norine, Harmonic morphisms and hyperelliptic graphs. International Mathematics Research Notices, 2009(15):2914–2955, 2009. https://doi.org/10.1093/imrn/rnp037
Lucia Caporaso, Algebraic and combinatorial Brill–Noether theory. In: Valery Alexeev, Angela Gibney, Elham Izadi, János Kollár, and Eduard Looijenga (editors), Compact moduli spaces and vector bundles, Contemporary Mathematics, pages 69–85. American Mathematical Society, 2012. https://bookstore.ams.org/conm-564
Scott Corry, Maximal harmonic group actions on finite graphs. Discrete Mathematics, 338(5):784–792, 2015. https://doi.org/10.1016/j.disc.2014.12.016

(The publication years of the 4th and 5th paper do not quite match the publication years listed in the lecture notes. However, the lecture notes do match the year that the corresponding preprints were first announced on arXiv.)

In addition to the papers 1–3 listed above, I consider the following papers to be essential reading on this topic:

Matthew Baker and Serguei Norine, Riemann–Roch and Abel–Jacobi theory on a finite graph, Advances in Mathematics, 215(2):766–788, 2007. https://doi.org/10.1016/j.aim.2007.04.012
Matthew Baker, Specialization of linear systems from curves to graphs, Algebra & Number Theory, 2(6):613–653, 2008. https://doi.org/10.2140/ant.2008.2.613
Filip Cools, Jan Draisma, Sam Payne, and Elina Robeva, A tropical proof of the Brill–Noether theorem, Advances in Mathematics, 230(2):759–776, 2012. https://doi.org/10.1016/j.aim.2012.02.019

You may also want to take a look at the 2016 survey by Baker and Jensen:

Matthew Baker and David Jensen, Degeneration of linear series from the tropical point of view and applications. In: Matthew Baker, Sam Payne (editors), Nonarchimedean and Tropical Geometry, Simons Symposia, pages 365–433, Springer, 2016. https://doi.org/10.1007/978-3-319-30945-3_11

Furthermore, I am aware of the existence of two textbooks on the more combinatorial side of things, but I'm not sure that these will contain what you are looking for:

Scott Corry and David Perkinson, Divisors and Sandpiles: An Introduction to Chip-Firing, American Mathematical Society, 2018.
Caroline J. Klivans, The Mathematics of Chip-Firing, Mathematical Association of America, 2018.

Finally, depending on your level, you might also enjoy the following "light reading":

Matt Baker, Riemann–Roch for graphs and applications, blog post, 2013. https://mattbaker.blog/2013/10/18/riemann-roch-for-graphs-and-applications/
Kevin Hartnett, Tinkertoy models produce new geometric insights, Quanta Magazine, 2018. https://www.quantamagazine.org/tinkertoy-models-produce-new-geometric-insights-20180905/
Jan Draisma and Alejandro Vargas, On the gonality of metric graphs, Notices of the American Mathematical Society, 68(5):687–695, 2021. https://doi.org/10.1090/noti2277
David Jensen, Chip firing and algebraic curves, Notices of the American Mathematical Society, 68(11):1875–1881, 2021. https://doi.org/10.1090/noti2378


To get an overview of recent developments in this field, I would suggest to start with the excellent expository article by Jensen (number 15 on the list), followed by the survey by Baker and Jensen (number 9 on the list). After that, either follow the references in those papers that you find interesting, or come back to list and take a look at some of the classics (1–3 and 6–8).