Spectral Graph Theory Lecture 7 Cheeger ’ s Inequality

Lecture 12 : Introduction to Spectral Graph Theory , Cheeger ’ s inequality

Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. Spectral graph theory seeks to relate the eigenvectors and eigenvalues of matrices corresponding to a Graph to the combinatorial properties of the graph. While generally the theorems are designed for unweighted and undirected graphs they can be extended to the weighted graph case and (less co...

MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lecture 20: Introduction to Spectral Graph Theory–III. Graph Cut and Cheeger Constants, Isospectral Graphs, and Discrete Laplacian Eigenvalue Problems

Example 1.2. In Figure 1, S is the set of dark nodes, and the dotted lines form ∂S. S ∆ = V \S. Also, vol(S) = m(S) ∆ = x∈S m(x).

Lecture 08 . RandomWalk on Graphs : Fiedler Vector , Cheeger inequality

In this class, we introduced the random walk on graphs. The last lecture shows Perron-Frobenius theory to the analysis of primary eigenvectors which is the stationary distribution. In this lecture we will study the second eigenvector. To analyze the properties of the graph, we construct two matrices: one is (unnormalized) graph Laplacian and the other is normalized graph Laplacian. In the first...

Cheeger Inequalities for Submodular Transformations

The Cheeger inequality for undirected graphs, which relates the conductance of an undirected graph and the second smallest eigenvalue of its normalized Laplacian, is a cornerstone of spectral graph theory. The Cheeger inequality has been extended to directed graphs and hypergraphs using normalized Laplacians for those, that are no longer linear but piecewise linear transformations. In this pape...

Lecture 03 : Chernoff Bounds and Intro to Spectral Graph Theory 3 1 . 1 Hoeffding ’ s Inequality