A few examples and counterexamples in spectral graph theory

Examples and Counterexamples in Graph Theory

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Spectral Theory Examples

Example 1 (Spectrum of Multiplication Operators) Let • (X, M, µ) be a σ–finite (actually semifinite will do) measure space, • 1 ≤ p ≤ ∞ and • a : X → C be a bounded measurable function on X. (Aϕ)(x) = a(x)ϕ(x) Point spectrum: Let λ ∈ C. Then λ1l − A is injective ⇐⇒ ϕ ∈ L p (X, M, µ), (λ − a(x))ϕ(x) = 0 a.e. =⇒ ϕ(x) = 0 a.e. ⇐⇒ λ − a(x) = 0 a.e.

Nonlinear Games: examples and counterexamples

In this session, popular nonlinear control methodologies are compared using benchmark examples generated with a "converse Hamilton-Jacobi-Bellman" method (CoHJB). Starting with the cost and optimal value function V, CoHJB solves HJB PDEs "backwards" algebraically to produce nonlinear dynamics and optimal controllers and disturbances. Although useless for design, it is great for generating bench...

Spectral graph theory

With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properies (the eigenvalues and eigenvectors) of these matrices pro...

Spectral graph theory