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Explore advanced techniques for asymptotically enumerating regular graphs in this fourth lecture from the PCMI graduate summer school series. Delve into three primary methodological approaches: 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. Focus particularly on the latter method while understanding how these enumeration techniques extend beyond mere counting to applications in generating graphs with specified degree sequences, network design and testing, network dynamics analysis, and real-world system modeling. Build upon foundational discrete probability theory concepts while examining concentration inequalities such as Chernoff and Azuma-Hoeffding bounds, with no prior knowledge of graph enumeration by degree sequence required. Engage with cutting-edge research in probabilistic and extremal combinatorics, understanding how these fields connect to analysis, geometry, number theory, statistical physics, and theoretical computer science through comprehensive mathematical exploration led by expert faculty from the Institute for Advanced Study's Park City Mathematics Institute.
