A few weeks ago, a few physicists (Peter Denton, Stephen Parke, and Xining Zhang) and one notable mathematician (Terence Tao) posted a short 3-page note to the arXiv, containing a “new” identity relating to eigenvectors to eigenvalues. Many people rushed to explain that it was not new; it has existed at least since 1966 and has been used many times by people who took no credit for it. The authors have since amended their paper and converted it into a survey containing all seven proofs of this identity, but not before it was picked up by Quanta. Now, in response to collaborators asking if I knew it and co-authors inquiring if I’ve thought about using it, I’m writing about how we usually arrive at identity in algebraic graph theory.

the adjacency matrix

Say we have a graph \(G\) on \(n\) vertices. The adjacency matrix \(A\) of \(G\) is an \(n\times n\) matrix with rows and columns indexed by the vertices of \(G\); \(A_{u,v} = 1\) if \(u\) is adjacent to \(v\) and \(A_{u,v} = 0\) otherwise. It may not be immediately apparent, but this matrix has the following property: \[ A^k_{u,v} = \text{the number of walks from } u \text{ to } v \text{ of length }k. \] We will write \(\phi(G,t)\) for the characteristic polynomial of \(A\).

the walk-generating function

Now we may consider the walk-generating function of \(u\) and \(v\) in \(G\), denoted \(W_{u,v}(G,t)\). This is a formal power series in variable \(t\), where the coefficient of \(t^k\) is the number of walks from \(u\) to \(v\) of length \(k\). But, oh wait, we already know how to count those! We have that \[ W_{u,v}(G,t) = \sum_{k=0}^{\infty} A^k_{u,v} t^k. \] If we look carefully at each term being summed, we are just extracting the \(u,v\) entry of some matrix and then multiplying by \(t^k\), so we can just do the extraction later and get: \[ W_{u,v}(G,t) = \sum_{k=0}^{\infty} (tA)^k_{u,v}. \] In fact, we could just take the whole sum first and then extraction the \(u,v\) entry and so \[ W_{u,v}(G,t) = \left(\sum_{k=0}^{\infty} (tA)^k\right)_{u,v}. \]

And now, we need to recall from calculus the following equality (keeping in mind that, since we work with these series as formal power series, we are not troubled by things like convergence): \[ 1 +x + x^2 + x^3 + \cdots = \frac{1}{1-x}. \] We can apply it to our expression for \(W_{u,v}(G,t)\) where \(x = tA\) and get \[ W_{u,v}(G,t) = (I-tA)^{-1}_{u,v}. \] While this looks a bit better, I’d much rather work with \(tI-A\), whose determinant is the characteristic polynomial of \(A\). So, we will do a change of variables as follows: \[ t^{-1}W_{u,v}(G,t^{-1}) = (tI-A)^{-1}_{u,v}. \] From this point forward, we will work with the expression \((tI-A)^{-1}_{u,v}\), with the knowledge that it gives the walk-generating function.

Cramer’s rule

We will use Cramer’s rule to simplify the diagonal entries of \((tI-A)^{-1}\); we can also do off-diagonal entries, see Godsil’s Algebraic Combinatorics but it involves a small amount of mess.

Cramer’s rule says that you can get the \(u,u\) entry of a matrix \(M\) by dividing the adjoint at \(u\) by the determinant. If you remove the \(u\)th row and the \(u\)th column from \(A\), you get an \(n-1\times n-1\) matrix which happens to be the adjacency matrix of \(G\) with the vertex \(u\) deleted, which we denote \(G\setminus u\). Thus \[ (tI-A)^{-1}_{u,u} = \frac{\phi(G\setminus u, t)}{\phi(G,t)}. \quad \quad \quad \quad (*) \] We will call this equation ([/latex]*[/latex]) mostly because I don’t know how to enable equation numbering in MathJax.

spectral decomposition

Now we will use spectral decomposition to obtain another expression for \((tI-A)^{-1}_{u,u}\). Since \(A\) is a real symmetric matrix, it is diagonalizable. Why is that so? It’s explained in a previous post. We may write \(A\) as follows: \[ A = \sum_{r=0}^d \theta_r E_r \] where the \(\theta\)s are the distinct eigenvalues of \(A\) and \(E_r\) is the idempotent projector for the \(\theta_r\)-eigenspace of \(A\). I always find this cool to use because we defined our matrix \(A\) to record adjacencies in \(G\), but now we are considering \(A\) as a matrix operator acting on the \(n\)-dimensional vector space; its action is entirely determined by how it acts on its eigenspaces, the subspaces on which it acts as scalar multiplication.

Recalling that eigenvectors of different eigenspaces are orthogonal, it’s not difficult to convince yourself that \[ A^2 = \sum_{r=0}^d \theta_r^2 E_r, \] where we notice that we only had to square the eigenvalues to square \(A\). In fact, \[ f(A) = \sum_{r=0}^d f(\theta_r) E_r, \] for all “reasonable” functions \(f\). We will not discuss the technical details of what it means to be reasonable, but it suffices to say that polynomial, rational functions and power series are all reasonable. In particular, \(\frac{1}{t-x}\) is reasonable and so \[ (tI-A)^{-1}_{u,u} = \sum_{r=0}^d \frac{1}{t- \theta_r} (E_r)_{u,u}. \quad \quad \quad (\dagger) \]

putting it all together

Putting together equation \((\dagger)\) and \((*)\), we get the following: \[ \frac{\phi(G\setminus u, t)}{\phi(G,t)} = \sum_{r=0}^d \frac{1}{t- \theta_r} (E_r)_{u,u}. \quad \quad (\dagger\dagger) \] You may complain that it’s not exactly what they have in their paper, but it’s actually just one short trick away.

Fix \(s \in \{0,\ldots,r\}\) and we will find the expression for \((E_s)_{u,u}\). We multiply everything by \(t-\theta_s\) to get \[\frac{(t-\theta_s)\phi(G\setminus u, t)}{\phi(G,t)} = \sum_{r\neq s} \frac{t-\theta_s}{t- \theta_r} (E_r)_{u,u} + (E_s)_{u,u}.\] Now we evaluate at \(t = \theta_s\) and get the desired eigen-magicks: \[ \frac{(t-\theta_s)\phi(G\setminus u, t)}{\phi(G,t)} |_{t = \theta_s} = (E_s)_{u,u}. \] You may think we need to take a limit; Gabriel Coutinho’s notes has a limit here, but it is, in fact, unnecessary. Suppose \(G\) has \(\theta_s\) as an eigenvalue with multiplicity \(m\). By the Interlacing Theorem, removing one vertex can only decrease the multiplicity by at most one and thus, there are also \(m\) factors of \(t-\theta_s\) appear on the top of the left hand side and everything cancels.

some uses of the eigen-magicks

I use equation \((\dagger\dagger)\) and the corresponding equation for the off-diagonal entries often; in the study of state transfers of quantum walks, we need to investigate something called strong cospectrality of pairs of vertices. In particular, in a paper with Chris Godsil, Mark Kempton and Gabor Lippner (which is cited by the new survey version of the paper by Denton et al.), we use these equations to perturb an edge of a strongly regular graph, so that the endpoints become strongly cospectral. For those interested, the arXiv version from 2017 is here and the JCTA version is here.

If you stare at equation \((\dagger)\) long enough, you may be tempted, as I was, to substitute \(t=0\) and obtain the entries of the inverse of \(A\). This is somewhat legitimate since, when \(A\) is invertible, \(t=0\) is not a pole of the right hand side. This is essential what I do for the class of invertible trees in a recent paper. I reprove a result of Godsil’s on the inverses of trees from the 80s, but I do it in such a way that I am able to define a combinatorial operation on a tree which strictly increases (or decreases, if you apply the inverse of the operation) the median eigenvalue of the tree. It’s surprising that you can do this; eigenvalues relate to combinatorial structure of the graph, but the relationship is, in general, not quid pro quod, as it is in this case. This gives a partial order on the class of trees with a perfect matching and I find the maximal and minimal elements. I leave its height and Moebius function as open problems for readers.

Some people make a habit of occasionally visiting Wikipedia’s front page and following hyperlinked phrases down a rabbit hole of articles. I do a similarly thing with Andries Brouwer’s tables of strongly regular graphs, which leads to a cascade of papers, checking BCN and computing stuff in Sage.  How does that happen exactly? As usual in math, we now need to start with some definitions.

Definitions

A strongly regular graph is a \(k\)-regular graph where every pair of adjacent vertices has \(a\) common neighbours and every pair of non-adjacent vertices has \(c\) common neighbours. The tuple \((n,k,a,c)\) is called the parameter set of the strongly regular graph (or SRG). For example, the Petersen graph is a SRG with parameter set \((10,3,0,1)\). We immediately, see that these number \(n,k,a\) and \(c\) have to satisfy some conditions, some trivial (\(n>k\) and \(a,c \leq k\) and others less so.

Brouwer’s tables consists of all possible parameter sets for \(n\) up to \(1300\) which have passed some elementary checks. The parameters which are in green are classes where it has been confirmed that there does in fact exist at least one SRG with those parameters. The ones in red have been shown by other means (failing to satisfy the absolute or relative bound, for example) to not be parameter sets of any SRGs. The entries in yellow are parameter sets where the existence of a SRG is still open. This leads us to …

Non-existence of SRGs

Finding whether or not a putative SRG exists is a pretty interesting problem; the existence of the \(57\)-regular Moore graph which is a SRG with parameters \((3250, 57,0,1)\) is a long-standing open problem in algebraic graph theory. Today, we’ll look at the parameter set \((49,16,3,6)\), which holds the distinction of being the first red entry in Brouwer’s tables which is not a conference graph and also not ruled out by the absolute bound. This means that it was ruled out by a somewhat ad-hoc method; in this case, it was ruled out by Bussemaker, Haemers, Mathon and Wilbrink in this 1989 paper. However, computers have come along way since 1989, so instead of following their proof, I will give another proof, which uses computation (which was done in Sage).

That’s way too many 2’s, this SRG can’t exist

Suppose \(X\) is a SRG with parameter set \((49,16,3,6)\). This graph has eigenvalues $$16, 2^{(32)}, \text{ and } -5^{(16)}$$ where the multiplicities are given as superscripts. We’ll proceed by trying to build \(X\). First, we “hang a graph by a vertex” as follows:

The main thing to note right away is that neighhourhood of a vertex \(x\) in \(X\), denoted \(\Gamma_1(x)\) in the diagram, is a \(3\)-regular graph on \(16\) vertices. This is excellent news because there are only 4060 such graphs and one can generate them using geng, or download them from Gordon Royle’s website.

Next, we apply interlacing. We denote by \(\lambda_i(G)\) the \(i\)th largest eigenvalue of \(G\). Since \(\Gamma_1(x)\) is an induced subgraph of \(X\), interlacing theorem tell us $$ \lambda_i(X) \geq \lambda_i(\Gamma_1(x)) \geq \lambda_{33 + i}(X)$$ for \(i=1,\ldots,16\). In particular, we see that the second largest eigenvalue of \(\Gamma_1(x)\) is at most  \(2\). In general, this gives us all kinds of useful information (for example, that \(\Gamma_1(x)\) has to be connected), but since we only care about cubic graphs on \(6\) vertices, we can just go ahead and check which of them have this property. In turns out that only 35 cubic graphs on 16 vertices have this property.

If I have an induced subgraph \(Y\) of \(X\) on 18 vertices, the  interlacing theorem would give the following:  $$ \lambda_2(X) \geq \lambda_2(\Gamma_1(x)) \geq \lambda_{33}(X).$$ Recalling that the second and 33rd largest eigenvalues of \(X\) are both \(2\), we see that any subgraph of \(X\) on 18 vertices must have its second eigenvalue equal to \(2\). Since \(\lambda_i(G)\) already has 16 vertices, we’re not far off. Let \(y\) be a vertex in the second neighbourhood of \(x\) and we will consider the subgraph \(Y\) induced by \(x\), the neighbours of \(x\) and \(y\).

In terms of the computation, we try to add \(x\) and \(y\) to each of the remaining 35 cubic graphs. We can always choose \(y\) to be adjacent to a fixed vertex in \(\Gamma_1(x)\) and so there are \({16 \choose 5}\) ways of adding the other \(5\) edges incident to \(y\). Thus we have 35(3003) candidates for \(Y\). A quick computation later, we see that none of those graphs have \(2\) as the second largest eigenvalue, which contradicts the existence of \(X\).

So what? And also, now what?

It’s fun proving the non-existence of a SRG that is already known not to exist, but clearly what we actually want to do is convert some yellow entries of Brouwer’s table to either red or green. The method here (as well as that of the original paper of  Bussemaker, Haemers, Mathon and Wilbrink) works because the multiplicity of \(2\) as an eigenvalue is really large. The degree of the graph is equal to the multiplicity of its smallest eigenvalue, which smells of some kind of duality. This happens, for example, in the complement of Clebsch graph (which, by the way, is formally self-dual). The next open case where this occurs is \((100,33,8,12)\).  In this case, the first neighbourhood of a vertex would be a \(8\)-regular graph on 33 vertices, with a fairly large spectral gap. We probably don’t want to be generating all such graphs, so some new idea is needed.

Some people believe in the statement: “if it’s not recreate-able, it’s not true.” … I don’t.
But instead of having dozens of photos all labelled “incomprehensible (but surely important) scribblings with math symbols”, I’m converting them into casual expositions. The posts will be generally aimed at an upper-year undergraduate level and will be about various topics in algebraic graph theory, like cores of cubelike graphs, generalized quadrangles and association schemes.