Eigenvalues of the Laplacian Matrix of the Complete Graph

We will show that the eigenvalues of the Laplacian matrix of the complete graph are where  the eigenvalue has (algebraic) multiplicity and the eigenvalue has multiplicity .

Spectral Sparsification by Spielman and Srivastava: How They Connected the Dots

In this post I will discuss my own understanding of the spectral sparsification paper by Daniel Spielman and Nikhil Srivastava (STOC 2008). I will assume the following:

1. The reader is a beginner, like me, and have already glanced through the Spielman-Srivastava paper (from now on, the SS paper).
2. The reader has, like me, a basic understanding of spectral sparsification and associated concepts of matrix analysis. I will assume that she has read and understood the Section 2 (Preliminaries) of the SS paper.
3. The reader holds a copy of the SS paper while reading my post.

First, I will mention the main theorems (actually, I will mention only what they “roughly” say).