Contents Chapter 1. Eigenvalues and the Laplacian of a graph 1 1.1. Introduction 1 1.2. The Laplacian and eigenvalues 2 1.3. Basic facts about the spectrum of a graph 6

We write M ∈ Rn×n to denote that M is an n×n matrix with real elements, and v ∈ Rn to denote that v is a vector of length n. Vectors are usually taken to be column vectors unless otherwise specified. Recall that a real matrix M ∈ Rn×n represents a linear operator from Rn to Rn. In other words, given any vector v ∈ Rn, we think of M as a function which maps it to another vector w ∈ Rn, namely th...

Theorem 1 (Arora, Rao, Vazirani, 2004) There is an O( √ log n)-approximation algorithm for sparsest cut. The proof of the theorem uses a SDP relaxation in terms of vectors vi ∈ Rn for all i ∈ V . Define distances to be d(i, j) ≡ ‖vi − vj‖ and balls to be B(i, r) ≡ {j ∈ V | d(i, j) ≤ r}. We first showed that if there exists a vertex i ∈ V such that |B(i, 1/4)| ≥ n/4, then we can find a cut of sp...

Let G^s be a signed graph, where G = (V;E) is the underlying simple graph and s : E(G) to {+, -} is the sign function on E(G). In this paper, we obtain k-th signed spectral moment and k-th signed Laplacian spectral moment of Gs together with coefficients of their signed characteristic polynomial and signed Laplacian characteristic polynomial are calculated.