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2. 1. Tuesday Lecture room 5203: (15:55-18:20);Thursday Lecture room 5204 : (14:00-15:35). |
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Lecture 1 Introduction; Harmonic functions and Laplacians on graphs. Lecture 2 Courant's minimax principle; Bipartite graphs . Lecture 3 Complete graphs; Diameter and eigenvalues; Alon-Boppana; Cylces; Eigenvalues of Cartesian products (I). Lecture 4 Eigenvalues of Cartesian products (II); Convergence to equilibrium: random walks and discrete-time heat equation; Cheeger constant. Lecture 5 Cheeger inequality; A brief comment for possible exteion to higher order inequalities; Qiao-Koolen-Markowsky conjecture: distance-regular graphs, dodecahedron and Hamming graphs; Dual Cheeger constant. Lecture 6 Dual Cheeger inequality; Relation between Cheeger and dual Cheeger constant; Buser inequality: Introduction; Bakry-Emery curvature. Lecture 7 Heat semigroup on finite graphs; Characterization of curvature bounds via gradient estimates; Proof of Buser inequality. Lecture 8 Curvature Matix: linear algebraic and functional viewpoints about Schur complement. Lecture 10 Curvature Matrix Examples: amply regular graphs; Johnson graphs revisited; locally Petersen graphs.. Lecture 11 Bakry-Emery curvature upper bounds: local and global connectivity. Lecture 12 Solving heat equation on infinite graphs: Dirichlet Laplacian and Maximal principle. Lecture 13 Solving heat equation on infinite graphs: Independence of the choices of exhaustions; Stochastic completeness. Lecture 14 Bakry-Emery curvature lower bounds; Chung-Yau Ricci flat graphs; Gradient estimate characterization. |
Homework 1 |
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1 Lecture notes for the course in 2021. 2 Qiao, Koolen, Markowsky, On the Cheeger constant for distance-regular graphs, Journal of Combinatorial Theory, Series A 173 (2020), 105227. 3 Horn, Purcilly and Stevens, Graph curvature and local discrepancy, J. Graph Theory 108 (2025), no. 2, 337-360. |
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