Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

Abstract

We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in Kennedy et al. (Calc Var PDE 60:61, 2021). These inequalities, which involve the first Betti number and the number of degree one vertices of the graph, recall both interlacing and other inequalities for the Laplacian eigenvalues of the whole graph, as well as estimates on the difference between the number of nodal and Neumann domains of the whole graph eigenfunctions. To this end we study carefully the principle of cutting a graph, in particular quantifying the size of a cut as a perturbation of the original graph via the notion of its rank. As a corollary we obtain an inequality between these energies and the actual Dirichlet and standard Laplacian eigenvalues, valid for all compact graphs, which complements a version for tree graphs of Friedlander’s inequalities between Dirichlet and Neumann eigenvalues of a domain. In some cases this results in better Laplacian eigenvalue estimates than those obtained previously via more direct methods.

Appendix A. Rigid partitions

In the current work we always considered general (“connected”) partitions of graphs, where cuts could be made anywhere, including in the middle of the partition elements (the clusters). This was natural for the theory developed: when creating partitions by cutting along eigenfunctions, as was central to the proofs of Theorems 1.1 and 1.2, a priori one has no control over where one is cutting.

On the other hand, inspired by the situation on domains, a second, more restricted class of partitions was also introduced in [21], where the graph can only be cut at the boundary of the partition clusters and not in their interior. This corresponds to the idea that when partitioning a domain $\Omega$ into pieces $\Omega _i$, there should be no “extra” incisions made in the interior of the $\Omega _i$; that is, $\partial \Omega _i$ should correspond exactly to where $\Omega _i$ meets the other $\Omega _j$. This second class may be defined as follows.

Definition A.1

We say a (connected) k-partition ${\mathcal {P}}=({\mathcal {G}}_1, \ldots , {\mathcal {G}}_k)$ of ${\mathcal {G}}$ is rigid if its boundary set $\partial {\mathcal {P}}$ (see Definition 2.12) coincides with the cut set ${\mathcal {C}}({\mathcal {G}}_{\mathcal {P}}:{\mathcal {G}})$ (see Definition 2.1). We denote the set of all exhaustive rigid k-partitions of ${\mathcal {G}}$ by ${\mathfrak {R}}_k = {\mathfrak {R}}_k ({\mathcal {G}})$.

Fig. 9

An example of a non-rigid 4-partition (right) of the graph from Fig. 1 (a) (reproduced here, left). The partition is not rigid due to the additional cut in the middle of the loop

The corresponding minimization problems were denoted as follows in [21, Section 4]:

$$\begin{aligned} {\mathcal {L}}^{N,r}_{k,p}({\mathcal {G}}) :={\left\{ \begin{array}{ll} \inf _{{{\mathcal {P}}}= ({\mathcal {G}}_1,\ldots ,{\mathcal {G}}_k) \in {\mathfrak {R}}_k}\left( \frac{1}{k}\sum \limits _{i=1}^k \mu _2({\mathcal {G}}_i)^p\right) ^{1/p} \qquad &{}\text {if } p \in (0,\infty ),\\ \inf _{{{\mathcal {P}}}= ({\mathcal {G}}_1,\ldots ,{\mathcal {G}}_k) \in {\mathfrak {R}}_k}\max \limits _{i=1,\ldots ,k} \mu _2({\mathcal {G}}_i) \qquad &{}\text {if } p = \infty , \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\mathcal {L}}^{N,c}_{k,p}({\mathcal {G}}) :={\left\{ \begin{array}{ll} \inf _{{{\mathcal {P}}}= ({\mathcal {G}}_1,\ldots ,{\mathcal {G}}_k) \in {\mathfrak {C}}_k}\left( \frac{1}{k}\sum \limits _{i=1}^k \mu _2({\mathcal {G}}_i)^p\right) ^{1/p} \qquad &{}\text {if } p \in (0,\infty ),\\ \inf _{{{\mathcal {P}}}= ({\mathcal {G}}_1,\ldots ,{\mathcal {G}}_k) \in {\mathfrak {C}}_k}\max \limits _{i=1,\ldots ,k} \mu _2({\mathcal {G}}_i) \qquad &{}\text {if } p = \infty . \end{array}\right. } \end{aligned}$$

Thus our ${{{\mathcal {L}}}^{N}_{k} ({\mathcal {G}})}$ equals ${\mathcal {L}}^{N,c}_{k,\infty }({\mathcal {G}})$ in [21]. Clearly ${\mathfrak {R}}_k \subset {\mathfrak {C}}_k$, whence ${\mathcal {L}}^{N,r}_{k,p}({\mathcal {G}}) \ge {\mathcal {L}}^{N,c}_{k,p}({\mathcal {G}})$ always (in the Dirichlet case, minimizing over ${\mathfrak {R}}_k$ or ${\mathfrak {C}}_k$ makes no difference [21, Lemma 4.3], justifying the common notation ${\mathcal {L}}^D_{k,p}({\mathcal {G}})$).

We briefly summarize how our Theorems 1.1 and 1.2 can be adapted to rigid partitions: as noted, since ${{\mathcal {L}}^{N,r}_{k,\infty }({\mathcal {G}}) \ge {{\mathcal {L}}}^{N}_{k} ({\mathcal {G}})}$, Theorem 1.1 can be obtained for free:

$$\begin{aligned} {\mathcal {L}}^{N,r}_{k,\infty }({\mathcal {G}}) \ge {\mathcal {L}}^{D}_{k+1-\beta }({\mathcal {G}}) \end{aligned}$$

for all $k \ge \max \{\beta ,1\}$. It is not at all clear whether Theorem 1.2 should hold in the rigid case; here, we merely observe that it will hold for k large enough since, by [18, Theorem 3.3], ${{\mathcal {L}}^{N,r}_{k,\infty }({\mathcal {G}}) = {{\mathcal {L}}}^{N}_{k} ({\mathcal {G}})}$ for sufficiently large $k \ge 1$ (how large depending on the graph): there exists some $k_0 = k_0 ({\mathcal {G}}) \ge \beta + |{\mathcal {V}}^1|$ such that

$$\begin{aligned} {\mathcal {L}}^{D}_{k}({\mathcal {G}}) \ge {\mathcal {L}}^{N,r}_{k+1-\beta -|{\mathcal {V}}^1|,\infty }({\mathcal {G}}) \end{aligned}$$

for all $k \ge k_0$. In principle it would be possible to estimate $k_0$ explicitly in terms of explicit features of the graph, but we expect the effort necessary to do so would be disproportionate to its value.