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Graphs with high second eigenvalue multiplicity

September 28, 2021

Milan Haiman,       Carl Schildkraut,       Shengtong Zhang

With Milan Haiman, Carl Schildkraut, and Shengtong Zhang (all undergrads at MIT), we just posted our new paper Graphs with high second eigenvalue multiplicity.

Question. How high can the second eigenvalue multiplicity of a connected bounded degree graph get?

In my earlier paper solving the equiangular lines problem (with fixed angle and high dimensions), we proved:

Theorem. (Jiang, Tidor, Yao, Zhang, and Zhao) A connected bounded-degree $n$-vertex graph has second eigenvalue multiplicity $O(n/\log\log n)$.

Here we are talking about eigenvalues of the adjacency matrix.

The fact that the second eigenvalue has sublinear multiplicity played a central role in our equiangular lines proof (as well as in our follow-up paper on spherical two-distance sets).

While the spectral gap (the difference between the top two eigenvalues) has been intensely studied due to the connections to graph expansion, not much is known about second eigenvalue multiplicities. The above theorem is the first result of this form about general classes of graphs.

Questions. Can the bound in the theorem be improved? What about when restricted to regular graphs, or Cayley graphs?

In particular, is $\le n^{1-c}$ true? We don’t know.

Why should we care? This appears to be a fundamental question in spectral graph theory. We saw that it is important for the equiangular lines problem (and maybe more applications yet to be discovered). In particular, improving the bounds in our theorem will improve the effectiveness of our equiangular lines theorem, which holds in “sufficiently high” dimension (how high is enough).

Eigenvalue multiplicities have already played an important role in Riemannian geometry, e.g., via connections to Gromov’s theorem on groups of polynomial growth; see the discussions in this paper by James Lee and Yury Makarychev.

Previously, the best example we knew has second eigenvalue multiplicity $\ge c n^{1/3}$. Basically any connected Cayley graph on \(\mathrm{PSL}(2, p)\) has this property, since a classic result of Frobenius tells us that every non-trivial irreducible representation of this group has dimension $\ge (p-1)/2$.

In our new paper, we construct several families of $n$-vertices, connected, and bounded degree graphs:

  • with second eigenvalue multiplicity $\ge c \sqrt{n/\log n}$;
  • Cayley graphs, with second eigenvalue multiplicity $\ge c n^{2/5}$;
  • with approximate second eigenvalue multiplicity $\ge c n/\log\log n$, thereby demonstrating a barrier to the moment/trace methods used in our earlier paper.

It would be interesting to find other constructions with second eigenvalue multiplicity exceeding $\sqrt{n}$. This appears to be a barrier for group representation theoretic methods, since every representation of an order $n$ group has dimension $\le \sqrt{n}$.

Another interesting recent paper by Theo McKenzie, Peter Rasmussen, and Nikhil Srivastava showed that connected bounded-degree $n$-vertex regular graphs have second eigenvalue multiplicity $\le n/(\log n)^c$, improving our $O(n/\log\log n)$ bound for general graphs. For general non-regular graphs, their result works for the normalized (random walk) adjacency matrix (rather than the usual adjacency matrix). Their result introduced the following novel observation: a closed walk of length $2k$ starting at an arbitrary vertex in a bounded degree graph typically covers $\ge k^c$ vertices.

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