Can One Hear the Spanning Trees of a Quantum Graph?

The question “Can one hear the shape of a quantum graph?” was answered in the affirmative by Gutkin and Smilansky GS01 for a quantum graph where the set of edge lengths are incommensurate. Their work was inspired by the famous question of Kac Kac66 which also sowed the seeds of results on isospectral billiards Cha95 ; GWW92 , graphs BPB09 and Riemannian manifolds Sun85 , along with many other related works. The connection between these questions is the extent to which properties of the spectrum can be used to recover information on the system’s geometry.

Isospectral domains were also studied in the context of discrete graphs Bro99 . In graph theory, a historic spectral geometric connection is provided by a theorem of Kirchhoff Kirchhoff . Kirchhoff’s matrix tree theorem expresses the number of spanning trees of a connected graph in terms of the spectral determinant of the combinatorial Laplacian of the graph which is the matrix $\mathbf{L}=\mathbf{D}-\mathbf{A}$ where $\mathbf{D}$ is a diagonal matrix of the vertex degrees and $\mathbf{A}$ is the adjacency matrix, see section 2.

In this article we reframe Kirchhoff’s theorem in the setting of quantum graphs. Quantum graphs were introduced to model electrons in organic molecules Pau36 and are widely employed in mathematical physics as a model of quantum mechanics in systems with complex geometry, see BKbook ; Kuc03 for a review. In a quantum graph the edges of the graph consist of intervals connected at the vertices with a self-adjoint differential operator on the set of intervals. Here we consider the most widely studied case of the Laplace operator with Neumann-Kirchhoff (or standard) vertex conditions, denoted $\mathcal{L}$. For Neumann-Kirchhoff conditions, functions on the graph are continuous and outgoing derivatives at the vertices sum to zero. We obtain the following relationship between the spectral determinant of $\mathcal{L}$ and the number of spanning trees of the graph.

If we have an equilateral quantum graph $\widetilde{\mathit{\Gamma}}$ where all the edge lengths are $\ell$, then $T_{\widetilde{\mathit{\Gamma}}}$ is precisely the number of spanning trees. We use $\widetilde{\mathcal{L}}$ to denote the Laplace operator of an equilateral quantum graph.

In the formula for $T_{\Gamma}$ (3), the number of edges $E$ can be determined from the spectrum of $\mathcal{L}$ via the Weyl law, where the mean spacing of the eigenvalues is given by $\pi^{-1}\sum_{e\in\mathcal{E}}l_{e}$, see BKbook , as the edge lengths are tightly constrained to $[\ell,\ell+\delta)$. For a connected graph $\Gamma$, the first Betti number is determined by the number of edges and vertices, $\beta=E-V+1$. Hence, it would be interesting to know if there are also spectral interpretations of the number of vertices and the product of the degrees in (3).

The article is organised as follows. In section 2 we introduce discrete graph and metric graph notation and operators. We relate the spectral determinants of equilateral quantum graphs and discrete graph operators in section 3. In section 4 we use Kirchoff’s matrix tree theorem and the relationship between spectral determinants to prove theorem 3. In section 5 we compare spectral determinants of equilateral and non-equilateral quantum graphs and in section 6 we prove theorem 2. Finally we summarize the results in section 7.

A discrete graph $G$ is comprised of a set of vertices $\mathcal{V}$ and a set of edges $\mathcal{E}$ such that each edge connects a pair of vertices; see for example Figure 1. Two vertices $u,v\in\mathcal{V}$ are adjacent if $(u,v)\in\mathcal{E}$ and we write $u\sim v$. We will denote the number of vertices by $V=\lvert\mathcal{V}\rvert$ and the number of edges by $E=\lvert\mathcal{E}\rvert$. The degree of a vertex $v$, denoted $d_{v}$, is the number of vertices that are adjacent to $v$. We let $\mathcal{E}_{v}$ denote the set of edges containing the vertex $v$ so $d_{v}=\lvert\mathcal{E}_{v}\rvert$. If $d_{v}=d$ for all $v\in\mathcal{V}$, the graph is regular. In this paper, we only consider simple graphs where there are no loops or multiple edges and the number of edges and vertices are finite. We also assume that $G$ is connected, so there is a path between every pair of vertices. The distance between a pair of vertices $u,v\in\mathcal{V}$ on a connected discrete graph is the number of edges in a shortest path joining $u$ and $v$. The diameter $D$ of the graph is the maximum distance between a pair of vertices. The first Betti number of $G$ is $\beta=E-V+1$, the number of independent cycles on $G$. A tree is a connected graph with no cycles, $\beta=0$. A spanning tree of $G$ is a subgraph of $G$ that is a tree and contains all the vertices of $G$.

A function on a discrete graph $G$ takes values at the vertices and therefore can be viewed as a vector $f\in\mathbb{C}^{V}$. The combinatorial Laplace operator, $\mathbf{L}$, is defined by

This means that we can define the self-adjoint $V\times V$ matrix $\mathbf{L}$ where

We note that $\mathbf{L}$ has zero as an eigenvalue (with multiplicity one for a connected graph), and we will index the eigenvalues of $\mathbf{L}$ in increasing order,

The harmonic Laplacian is

where $\mathbf{D}$ is a diagonal matrix of the degrees, $\mathbf{D}_{uv}=d_{v}\delta_{uv}$. We denote the eigenvalues of $\mathbf{\Delta}$ with $\lambda_{j}$ for $j=1,\dots,V$.

A metric graph $\mathit{\Gamma}$ is a discrete graph where every edge $e\in\mathcal{E}$ is assigned a length $\ell_{e}>0$. We associate the edge $e$ with the interval $[0,\ell_{e}]$. For $e=(u,v)$, we set $x_{e}=0$ at $u$ and $x_{e}=\ell_{e}$ at $v$; the choice of orientation of the coordinate is arbitrary and will not affect the results. The total length of a metric graph is the sum of the lengths of each edge, $\ell_{\mathrm{tot}}=\sum_{e\in\mathcal{E}}\ell_{e}$. An equilateral metric graph $\widetilde{\mathit{\Gamma}}$ is a metric graph where all the edge lengths are equal. We will use the term generic metric graph to distinguish a non-equilateral graph. A function $f$ on $\mathit{\Gamma}$ is a collection of functions $\{f_{e}\}_{e\in\mathcal{E}}$ where $f_{e}$ is a function on the interval $[0,\ell_{e}]$.

A quantum graph is a metric graph $\mathit{\Gamma}$ along with a self-adjoint differential operator. In this paper, we consider the Laplace operator, denoted by $\mathcal{L}$, which acts as $-\frac{\mathrm{d}^{2}}{\mathrm{d}x_{e}^{2}}$ on functions defined on $[0,\ell_{e}]$. We will use the notation $\widetilde{\mathcal{L}}$ if $\widetilde{\mathit{\Gamma}}$ is an equilateral metric graph. At the vertices, functions will satisfy the Neumann-Kirchhoff (or standard) vertex conditions,

By convention, $\dfrac{\mathrm{d}f_{e}}{\mathrm{d}x_{e}}(v)$ is taken to be the outgoing derivative at the end of the interval $[0,\ell_{e}]$ corresponding to $v$. The second Sobolev space on $\mathit{\Gamma}$ is the direct sum of the second Sobolev spaces on the set of intervals,

The domain of $\mathcal{L}$ is the set of functions $f\in H^{2}(\mathit{\Gamma})$ that satisfy (9). With these vertex conditions, $\mathcal{L}$ has an infinite number of non-negative eigenvalues BKbook which we will denote as

with $k_{j}\in\mathbb{R}$.

Some of these eigenvalues may correspond to eigenfunctions where $f_{e}(0)=f_{e}(\ell_{e})=0$ for all $e\in\mathcal{E}$. In this case the eigenfunction is also an eigenfunction of the Laplacian with Dirichlet vertex conditions. The spectrum of the graph with Dirichlet vertex conditions is just the union of the spectra of the Laplacian on $E$ disconnected intervals $[0,\ell_{e}]$ with Dirichlet boundary conditions. We call this the Dirichlet spectrum of $\mathit{\Gamma}$.

In this section we relate the spectral determinants of $\mathbf{\Delta}$ and $\widetilde{\mathcal{L}}$. This is used in section 4 to prove theorem 3. The spectral determinant of the harmonic Laplacian $\mathbf{\Delta}$ is the product of its nonzero eigenvalues,

The prime shows that eigenvalues of zero are omitted from the product. Correspondingly, the spectral determinant of the Laplace operator $\widetilde{\mathcal{L}}$ on an equilateral metric graph is, formally,

The spectral determinant of quantum graphs has been studied in a number of situations Des00 ; Des01 ; Fri06 ; HarKir11 ; HarKirTex12 . Here we use a zeta function regularization of the product. The spectral zeta function, $\mathcal{Z}(s)$, is a generalization of the Riemann zeta function where the nonzero eigenvalues take the place of the integers. The spectral zeta function corresponding to $\widetilde{\mathcal{L}}$ is

which converges for $\operatorname{Re}(s)>1$. Making an analytic continuation to the left of $\operatorname{Re}(s)=1$, the regularized spectral determinant is defined as

There is a correspondence between the eigenvalues of $\mathbf{\Delta}$ and $\widetilde{\mathcal{L}}$ Below85 (see also Kuc03 ; Pank06 ; BKbook ).

By proposition 1, every eigenvalue of $\mathbf{\Delta}$ can be written as $1-\cos(t\ell)$ for some $t\in\left[0,\frac{\pi}{\ell}\right]$. Let $T=\{t_{j}\}_{j=1}^{V}$ be the set $0=t_{1}<t_{2}\leq\ldots\leq t_{V}\leq\frac{\pi}{\ell}$ such that $1-\cos(t_{j}\ell)\in\sigma(\mathbf{\Delta})$ (including multiplicity). We know $0=t_{1}<t_{2}$ as zero is an eigenvalue of $\mathbf{\Delta}$ with multiplicity one when $G$ is connected. This allows us to write the spectral zeta function of $\widetilde{\mathcal{L}}$ in terms of the finite set $T$. The following lemma is a combination of equations (20) and (24) from HarWey18 .

Using lemma 1 we can obtain a relationship between the spectral determinants of the harmonic Laplacian of a discrete graph and the Laplacian of the corresponding equilateral quantum graph.

Since $\zeta_{H}(0,a)=\frac{1}{2}-a$ NITS , it follows that,

Using lemma 1,

since $\zeta_{R}(0)=-1/2$, $\zeta^{\prime}_{R}(0)=-\ln(2\pi)/2$, and $\zeta^{\prime}_{H}(0,a)=\ln(\Gamma(a))-\frac{1}{2}\ln(2\pi)$ NITS . Additionally $\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)$ NITS and $4\sin^{2}\left(\frac{t_{j}\ell}{2}\right)=2(1-\cos(t_{j}\ell))=2\lambda_{j}$, so

Taking the exponential of this expression yields the result.

Kirchhoff’s matrix tree theorem Kirchhoff gives the number of spanning trees of a discrete graph in terms of the spectral determinant of the combinatorial Laplacian. In this section we prove theorem 3 which reframes this result in terms of the spectral determinant of an equilateral quantum graph.

In this section we compare the spectral determinants of equilateral and generic quantum graphs where the corresponding discrete graphs are the same. In particular, given an equilateral graph $\widetilde{\mathit{\Gamma}}$ with edge length $\ell$, we will consider a generic graph whose edge lengths lie in the interval $[\ell,\ell+\delta)$. Friedlander computed the spectral determinant of a generic quantum graph Fri06 .

First, we note that the proof of lemma 2 also showed

In the case of an equilateral graph with edge length $\ell$, we will define $\widetilde{\mathbf{R}}=\ell^{-1}\mathbf{L}$, which agrees with (41) when $\ell_{e}=\ell$ for every edge $e$. In fact, setting the edge lengths equal in theorem 4,

where we used proposition 2 for the last step. This demonstrates that evaluating (40) with equal edge lengths produces the spectral determinant of the equilateral graph as expected.

For a generic quantum graph $\mathit{\Gamma}$ with edge lengths in $[\ell,\ell+\delta)$, let

Notice that, by theorem 3, $T_{\widetilde{\mathit{\Gamma}}}$ is the number of spanning trees of an equilateral graph. For $\delta$ sufficiently small, the value of $T_{\mathit{\Gamma}}$ will be close enough to $T_{\widetilde{\mathit{\Gamma}}}$ to determine the number of spanning trees of a generic quantum graph. Using theorem 4,

and similarly,

We will determine a bound on $\delta$ such that $|T_{\mathit{\Gamma}}-T_{\widetilde{\mathit{\Gamma}}}\rvert<1/2$, and hence the number of spanning trees is the closest integer to $T_{\mathit{\Gamma}}$. To this end, we employ two bounds on the spectrum of $\mathbf{L}$. The first is a lower bound on the second smallest eigenvalue in terms of the graph diameter $D$, the maximum distance between a pair of vertices, due to McKay Moh91 .

We can conclude from theorem 5 that $\tilde{\lambda}_{2}>4/V^{2}\ell$, and we know by applying theorem 6 that ${\det}^{\prime}(\mathbf{L})\leq V^{V-1}$. Using these inequalities we see that

which establishes theorem 2.

In this paper, we proved an analog of Kirchhoff’s matrix tree theorem for quantum graphs. In particular, we determined the number of spanning trees of an equilateral quantum graph from its spectral determinant. To do this we related the spectral determinant of an equilateral quantum graph to the spectral determinant of the harmonic Laplacian of the corresponding discrete graph. We extended this to non-equilateral quantum graphs where the edge lengths are sufficiently constrained.

The bound on the permitted variance in the edge lengths in theorem 2 is suboptimal but requires minimal information on the structure of the graph. However, the constraint may be loosened in some situations, even if there is no additional information about the graph’s structure. For example, all eigenvalues of $\mathbf{L}$ satisfy AndMor ,

As we assume in theorem 2 that we know the product of the vertex degrees, if we also know that $\mbox{max}_{(u,v)\in\mathcal{E}}(d_{u}+d_{v})<V$, then we can use (68) in place of theorem 6 which weakens the constraint on the spread of the edge lengths.

Acknowledgments

The authors would like to thank Gregory Berkolaiko for helpful comments. This work was partially supported by a grant from the Simons Foundation (354583 to Jonathan Harrison).