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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.

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  • $\begingroup$ See some references in my answer here. $\endgroup$
    – Moishe Kohan
    Commented Nov 22, 2021 at 17:18
  • $\begingroup$ Welcome to MathOverflow! To improve your question, could you maybe be a bit more specific? Do you have a link to the referenced lecture, so that we don't have to guess its contents based on the title? $\endgroup$
    – J. van Dobben de Bruyn
    Commented Nov 22, 2021 at 20:15

3 Answers 3

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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:

  1. 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
  2. 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
  3. 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
  4. 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
  5. 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:

  1. 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
  2. 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
  3. 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:

  1. 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:

  1. Scott Corry and David Perkinson, Divisors and Sandpiles: An Introduction to Chip-Firing, American Mathematical Society, 2018.
  2. 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":

  1. Matt Baker, Riemann–Roch for graphs and applications, blog post, 2013. https://mattbaker.blog/2013/10/18/riemann-roch-for-graphs-and-applications/
  2. Kevin Hartnett, Tinkertoy models produce new geometric insights, Quanta Magazine, 2018. https://www.quantamagazine.org/tinkertoy-models-produce-new-geometric-insights-20180905/
  3. 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
  4. 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).

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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.

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  • $\begingroup$ Users are always apologizing for "self-promotion". Why? If you truly believe the ideas you promote through publications can be of little or no interest to others, don't publish/blog, but the quintessence of intellectual progress is sharing ideas. No need to apologize unless you are being intentionally or inadvertently toxic and wish to repent otherwise it comes across as false modesty or, worse, discourages others to offer up their own notes. $\endgroup$
    – Tom Copeland
    Commented Nov 22, 2021 at 19:33
  • $\begingroup$ The absurd logical conclusion is to not put your name on your papers. (Then you can avoid all criticism.) $\endgroup$
    – Tom Copeland
    Commented Nov 22, 2021 at 19:42
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    $\begingroup$ @TomCopeland It is usually regarded as impolite and inappropriate to brag about one's results. Apologising for self-promotion is a way to acknowledge this and advertise the quote is not to be understood as bragging, or led by vanity. It can read like “Some people enjoy quoting themselves, I do not.” Many authors prefer to not repeat their names in a paper they write, just mentioning “the author(s)”. $\endgroup$
    – Michaël Le Barbier
    Commented Nov 22, 2021 at 20:41
  • $\begingroup$ To offer one's notes to help others understand a topic and perhaps solve problems and owning up to it is not bragging by any definition--that's the point of Q&A sites and other types of collaboration. To say "It has been widely cited by subsequent papers on the subject" is, perhaps, so following your sentiments maybe a "I hate to brag, but it has been widely ... " would be demanded by your support of the gesturing. $\endgroup$
    – Tom Copeland
    Commented Nov 22, 2021 at 22:27
  • $\begingroup$ (cont) Nevertheless, it encourages following up on the paper and I think empty apologetic posturing is unnecessary. (Btw, I've always found Roland's and his colleagues work to be very interesting and he should be understandably proud about it.) $\endgroup$
    – Tom Copeland
    Commented Nov 22, 2021 at 22:28
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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.

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    $\begingroup$ I interpreted the original question differently: There is quite an activity around graphs as a sort of 'toy models" for Riemann surfaces. They are thus not considered as embedded in Riemann surfaces (the topic of the book, I guess) but as sort of 'trivial analogues'. $\endgroup$
    – Roland Bacher
    Commented Nov 22, 2021 at 17:04
  • $\begingroup$ @RolandBacher: Couldn't quickly find the lecture the OP mentioned, so I'm not sure precisely what he is looking for. $\endgroup$
    – Tom Copeland
    Commented Nov 22, 2021 at 17:09
  • $\begingroup$ Neither could I but Mednykh seems to have papers on this 'toy-model' stuff. $\endgroup$
    – Roland Bacher
    Commented Nov 22, 2021 at 18:40
  • $\begingroup$ And graph theory pops up everywhere. In any event, it's an interesting book on such topics that might be of interest to others browsing the question. $\endgroup$
    – Tom Copeland
    Commented Nov 22, 2021 at 18:50
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    $\begingroup$ I believe this is the "correct" answer, in the sense that this book gives a broad overview of the field Mednykh seems to typically work in, e.g. sciencedirect.com/science/article/pii/S0195669803001379 $\endgroup$
    – Jacob Bond
    Commented Nov 23, 2021 at 6:38

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