I’m leading a directed study with a student about spectral graph theory this semester. (Long story short, he was in my linear algebra class, another student saw a cool thing about clustering using eigenvalues of the graph Laplacian in Tim Chartier’s book, and the three of us did some summer research work about it.) This means that I am learning a lot of stuff about graph theory and linear algebra, and this often bleeds out into other weird branches of mathematics. Here’s a good example, the moral of which is that if something is hard or annoying, you may be looking at it wrong.

Suppose that $G$ is a graph with vertex set $V(G)$ and edge set $E(G)$. Build the degree matrix $D$ (a diagonal matrix indexed by the vertices whose diagonal entries are given by the degree of each vertex) and the adjacency matrix $A$ (a matrix indexed by the vertices such that $A(v,u) = 1$ if $v$ is adjacent to $u$ and $0$ otherwise). The graph Laplacian is the matrix $L = D - A$.

Pick some function $f:V(G) \to \Reals$. It’s useful to think of $f$ as a column vector in $\Reals^{\vert V \vert}$. The Rayleigh quotient of $f$ with respect to the Laplacian is defined as:

\[\frac{\langle f, Lf \rangle}{\langle f, f \rangle}\]

The notes we were looking at assert the following:1

\[\frac{\langle f, Lf \rangle}{\langle f, f \rangle} = \frac{\displaystyle\sum_{(u, v) \in E}(f(u)-f(v))^2}{\displaystyle\sum_v f(v)^2 d_v}\]

We wanted to better understand this calculation, because the Rayleigh quotient is important for picking out eigenvalues of \(L\). The denominator ends up being straightforward, but in the numerator, I dragged us lengthwise through the following honestly horrific calculation, which you should not look very hard at other than to note that it involves double sums, index gymnastics, overly-clever relabeling, and every other bad thing.

\[\begin{align*} \langle f, Lf \rangle &= \sum_u f(u) (Lf(u)) = \sum_u f(u) \left[\sum_v L(u, v) f(v)\right] \\ & = \sum_u \sum_v f(u) (D(u,v) - A(u,v)) f(v) \\ &= \sum_u \left[ d_u f(u)^2 - \sum_{v \text{ adj. } u}f(u) f(v) \right] = \frac12 \left( \sum_u \left[ 2 d_u f(u)^2 - \sum_{v \text{ adj. } u}2 f(u) f(v) \right] \right) \\ &= \frac12 \left( \sum_u d_u f(u)^2 + \sum_u d_u f(u)^2 - \sum_u\sum_{v \text{ adj. } u}2 f(u) f(v) \right) \\ &= \frac12 \left( \sum_u d_u f(u)^2 + \sum_v d_v f(v)^2 - \sum_u\sum_{v \text{ adj. } u}2 f(u) f(v) \right) \\ &= \frac12 \left( \sum_u \left[\sum_{v\text{ adj. } u} 1\right] f(u)^2 + \sum_v \left[\sum_{u\text{ adj. } v} 1\right] f(v)^2 - \sum_u\sum_{v \text{ adj. } u}2 f(u) f(v) \right) \\ &= \frac12 \sum_u \sum_{v\text{ adj. }u} f(u)^2 + \frac12 \sum_v \sum_{u\text{ adj. }u} f(v)^2 - \frac12 \sum_u\sum_{v \text{ adj. }u} 2f(u) f(v) \\ &= \sum_{(u, v) \in E} f(u)^2 + \sum_{(u, v) \in E} f(v)^2 - \sum_{(u, v) \in E} 2f(u) f(v) \\ &= \sum_{(u, v) \in E} \left[f(u)^2 + f(v)^2 - 2 f(u) f(v) \right] = \sum_{(u, v) \in E}(f(u) - f(v))^2 \end{align*}\]

Any time a calculation turns out this terrible and requires this many weird tricks, that’s a good sign that the approach you have chosen is somehow, morally, the wrong choice. So I went fishing on math twitter.

(Please ignore the typos, lol.)

The consensus of a couple of my smart twitter pals was that you have better luck if you think of $L$ not as $D-A$ – which was, ultimately, the choice that led to every other bad thing in that calculation above – but instead as $B B^T$. But who is this mysterious $B$, you ask? Good question.

$B$ is the incidence matrix. The rows of $B$ are indexed by the vertices of $G$, and the columns of $B$ are indexed by the edges of $G$. Put some kind of orientation on all the edges; I don’t care what it is. Then:

\[B(v, e) = \begin{cases} 1 & e \text{ starts at } v \\ -1 & e \text{ ends at } v \\ 0 & \text{otherwise} \end{cases}\]

(Note, btw, that I don’t care if you choose the opposite sign convention for $B$; everything still works out the same.)

You should spend one to four minutes convincing yourself that $B B^T = D - A$; just kinda imagine what happens on each row. Once you’ve done that, you can think about $B$ itself. If you know what homology is (which, uh, maybe I do sometimes?), you may be thinking to yourself, huh, $B$ is some kind of linear map $B:E(G) \to V(G)$ to where $B(e) = u -v$, gee whiz, that sounds like the boundary operator. You’re right! What’s more, $B^T$ is therefore the coboundary operator. After all, a graph is nothing more than the 1-skeleton of a simplicial complex, so we may as well do homology stuff to it.

One more kinda complication here is that we started with a function $f:V(G) \to \Reals$ – that is, a vertex form – which we wanted to think about as $f\in\Reals^V$. Maybe we actually want to do some de Rham cohomology, then. That’s fine; $B^T$ is still the coboundary operator. Specifically, $B^Tf(u, v) = f(u) - f(v)$. You can think of $B^T:\Reals^V\to\Reals^E$ or you can think of $B^T:\Omega^0(G) \to \Omega^1(G)$, whichever you like more.

Here is the punchline of this blog post. With this setup, the Rayleigh quotient calculation becomes a one-liner:

\[\langle f, Lf \rangle = \langle f, B B^T f \rangle = \langle B^T f, B^T f \rangle = \sum_{(u, v) \in E} (B^T f(u,v))^2 = \sum_{(u, v) \in E} (f(u) - f(v))^2.\]

And if you don’t think that kicks ass, then you can get the hell out of my face.

  1. NB that things are set up a little differently there, using the normalized Laplacian, but this Rayleigh quotient calculation works out the same way. ↩