As pointed out by others, itIt seems that the question is not clear whatabout the contents of this lecture are.
However, after a quick internet search I did find some older (2018)following lecture notes with the same title written by the same professor: http://math.nsc.ru/conference/g2/g2r2/files/pdf/Lecture-8.pdf
In addition to the papers 1–3 listed above, I consider the following papers to be essential reading on this topic:
- 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://www.ams.org/journals/notices/202105/rnoti-p687.pdfhttps://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).