Remarks on Lecture 2

In this lecture, we generalized our setting to weighted graphs {G=(V,E,w)} where {w : E \to \mathbb R^+} is a positive weight function on edges. We also looked at the basic spectral theory of the normalized Laplacian, defined by

{\mathcal L_G = I - D_G^{-1/2} A_G D_G^{-1/2}} ,

which operates on functions {f : V \to \mathbb R} via

{\displaystyle \mathcal L_G f(u) = f(u) - \sum_{v \sim u} \frac{w(u,v)}{\sqrt{w(u) w(v)}} f(v)} .

In particular, we associated with {\mathcal L_G} the normalized Rayleigh quotient,

{\displaystyle \mathcal R_G(f) = \frac{\sum_{u \sim v} w(u,v) (f(u)-f(v))^2}{\sum_{u \in V} w(u) f(u)^2}} ,

and we showed that we can express the second-smallest eigenvalue as

{\lambda_2(\mathcal L_G) = \min \{ \mathcal R_G(f) : \sum_{v \in V} w(v) f(v) = 0 \}} .

We defined the expansion of a subset {S \subseteq V} by

{\phi_G(S) = \mathcal R_G(\mathbf{1}_S)} ,

where {\mathbf{1}_S} is the characteristic function for {S} .

Finally, we defined the expansion of a graph by

{\Phi_G = \min \left\{ \phi_G(S) : 0 < w(S) < w(V)/2\right\}}

We then stated and (almost) proved the following discrete Cheeger inequality.

Theorem 1. For any weighted graph {G=(V,E,w)} , we have

{\displaystyle \frac{\lambda_2}{2} \leq \Phi_G \leq 2 \sqrt{\lambda_2}} ,

where {\lambda_2 = \lambda_2(\mathcal L_G)} .

Exercises (optional)

  1. Prove that for the normalized Laplacian, just as for the combinatorial Laplacian, the number of connected components in a graph G is precisely the multiplicity of the smallest eigenvalue.
  2. We saw the the right-hand inequality in Theorem 1 is asymptotically tight. Prove that the left-hand side is asymptotically tight for the complete graph on n nodes.
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Remarks on Lecture 1

Today we started studying the spectrum of the combinatorial Laplacian on a graph G=(V,E) . This is the operator L_G : \mathbb R^V \to \mathbb R^V given by

{\displaystyle L_G f(x) = \mathrm{deg}(x) f(x) - \sum_{y \sim x} f(y)\,.}

We saw that L_G has a spectrum of the form 0 = \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n , and that the number of connected components in G is precisely the multiplicity of \lambda_1 .

I encourage you to look at the first four lectures of Spielman’s course for a refresher.

Exercises

The best way to learn a subject is to get your hands dirty. Here are some optional exercises to help build your intuition.

Recall that for a function f : V \to \mathbb R , we define its Rayleigh quotient by

\displaystyle \mathcal R_G(f) = \frac{\sum_{x \sim y} (f(x)-f(y))^2}{\sum_{x \in V} f(x)^2}.

If P_n is the path graph on n vertices, then as we argued today, we have \lambda_k \asymp \left(\frac{k}{n}\right)^2  .

  1. Prove that \lambda_2 \asymp \frac{1}{n^2}  by showing that (a) there exists a map f : V \to \mathbb R  with f \perp \mathbf{1} and \mathcal R_G(f) \lesssim \frac{1}{n^2} and (b) for any such map with f \perp \mathbf{1} , we have \mathcal R_G(f) \gtrsim \frac{1}{n^2} . We did this in class, but it will help to reconstruct the proof on your own.

  2. Try to prove that \lambda_k \lesssim \left(\frac{k}{n}\right)^2 by exhibiting an explicit subspace of test functions
    that achieves this bound. It may help to use the Courant-Fischer min-max principle
    which says that

    \displaystyle \lambda_k = \min_{S \subseteq \mathbb R^V} \max_{0 \neq f \in S} \mathcal R_G(f),

    where the minimum is over all k-dimensional subspaces S .

    [Hint: Have your subspace be the span of k functions with disjoint supports, where the support of a function f : V \to \mathbb R  is \mathrm{supp}(f) = \{ x \in V : f(x) \neq 0 \} .]

Welcome

This is the course web site for CSE 599S: Algorithmic Spectral Graph Theory. This is where notes, homeworks, and video will be posted.

The class will be held every Monday and Wednesday, from 3-4:20pm in CSE 305.