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Explore advanced techniques for asymptotically enumerating regular graphs in this graduate-level mathematics lecture from the IAS/PCMI Park City Mathematics Institute. Learn three primary enumeration methods: the configuration model and switchings, generating functions combined with saddle point methods, and the relatively recent Stein's method approach utilizing degree switchings to derive recursive formulas. Discover how these enumeration techniques extend beyond mere counting to applications in generating graphs with specified degree sequences, network design and testing, understanding network dynamics, and modeling real-world systems. Examine the mathematical foundations underlying each approach with particular emphasis on the Stein's method framework. Gain insights into how regular graph enumeration connects to broader themes in extremal and probabilistic combinatorics, including applications to analysis, geometry, number theory, statistical physics, and theoretical computer science. The presentation assumes familiarity with basic discrete probability theory and benefits from knowledge of concentration inequalities such as Chernoff and Azuma-Hoeffding bounds, though no prior experience with graph enumeration by degree sequence is required.
