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Explore advanced asymptotic enumeration techniques for regular graphs in this fifth 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 innovative Stein's method utilizing degree switchings for recursive formulas. Master the theoretical foundations and practical applications of regular graph enumeration, which extends beyond mere counting to include generating graphs with specified degree sequences, network design optimization, testing protocols, and real-world system modeling. Examine how these enumeration techniques contribute to understanding network dynamics and provide computational tools for complex combinatorial problems. Build upon prerequisite knowledge of discrete probability theory while gaining exposure to concentration inequalities such as Chernoff and Azuma-Hoeffding bounds, with no prior graph enumeration experience required. Engage with cutting-edge research at the intersection of extremal and probabilistic combinatorics, connecting discrete mathematics with analysis, geometry, number theory, statistical physics, and theoretical computer science through rigorous mathematical frameworks and contemporary applications.
