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Explore graphs, Hurwitz numbers, and commuting operators in this 56-minute conference talk by Aleksandr Yur'evich Orlov at the Centre de recherches mathématiques (CRM). Delve into the fascinating world of graphs and generalized cut-and-join relations, leading to a conjecture about families of commuting multivariable differential operators. Part of the Workshop on the Role of Integrable Systems dedicated to John Harnad, this presentation offers valuable insights into advanced mathematical concepts and their interconnections.