2.

1. Tuesday Lecture room 5203: (15:55-18:20);Thursday Lecture room 5204 : (14:00-15:35).

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 9 Curvature Matrix Examples: complete graphs, cycles, lattice graphs, hypercubes, regular trees, complete bipartite graphs, crown graphs, Johnson graphs..

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.

Lecture 15 Application of gradient estimate characterization: Stochastic completeness and diameter bound; Diameter bound via Lin-Lu-Yau curvature.

Lecture 16 Diameter of graphs with some negative Bakry-Emery curvature: Perpectual cutoff semigroups.

Lecture 17 Time derivative of perpectual cutoff semigroups; Gradient estimates; diameter bounds with cuvature bounds outside of a subset.

Lecture 18 Discussions on open problems about perpectual cutoff semigroup; Diameter bound under CD(K,n) condition.

Lecture 19 Chung-Lin-Yau eigenvalue-diameter-curvature estimate and recent improvement by Meng and Lin.

Lecture 20 Finite Markov Chains; The logarithmic mean; discrete vector analysis; Entropic Ricci curvature; A modified Gamma calculus.

Lecture 21 Gradient estimate characterization; The first nonzero eigenvalue as curvature value; Lichnerowicz estimate with non-constant curvature bound.

Lecture 22 Otto's calculus: tangent spaces; Riemmanian metric; geodesic equations; Hessian of entropy.

Lecture 23 Discrete Jordon-Kinderlehrer-Otto theorem; Diameter estimate under condition CD_\theta(K,N).

Lecture 24 Localized entropic curvature: A local gradient estimate and the reverse Poincare inequality.

Lecture 25 No expanders with nonnegative curvature.

Lecture 26 Eigenfunctions of graphs: cycles, hypercubes, stars, Petersen graphs; Strong and weak nodal domains; Discrete nodal domain theorem (I).

Lecture 27 Discrete nodal domain theorem (II); Nodal geography.

Lecture 28 Strong nodal domain estimates due to Lin-Lippner-Mangoubi-Yau.

Lecture 29 Signed graphs: balance, switching, and Laplacian.

Lecture 30 Cheeger inequality for signed graphs; Curvature of signed graphs.

Lecture 31 Gradient estimate and diameter bounds for eigenvalues of signed graphs (Wei Chen).

Homework 1

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.