Spectral minimal partitions of combinatorial graphs

Abstract: This paper investigates spectral minimal partitions for weighted graphs, thus extending the extensive class of results that are currently available on domains and, to a lesser extent, manifolds and metric graphs. We provide a rigorous framework for analyzing graph Laplacians under Dirichlet, Neumann, and boundaryless energy formulations; a central focus of the study is establishing existence theorems for minimal partitions. While existence is straightforward for finite connected graphs due to the finiteness of the class of admissible partitions, infinite graphs require advanced topological and functional-analytic machinery. Specifically, we introduce the notion of canonical compactifiability, which relates to compact embeddings and uniform Poincaré-type constants for Neumann and boundaryless energies; and an appropriate notion of subgraph convergence. In this way, we can relax the spectral minimal problem on infinite graphs by reducing it to the study of finite graphs; and can, thus, guarantee that optimal spectral energies are actually attained by appropriate partitions even in non-compact settings.

• The paper develops an existence theory for spectral minimal partitions on both finite and infinite weighted combinatorial graphs, focusing on three Laplacian realisations: Dirichlet, Neumann, and boundaryless.
• Key results include the determination of existence conditions for minimizers under different Laplacian realisations and the derivation of quantitative energy bounds, demonstrating the impact of boundary effects and non-contractible paths.
• The paper also highlights that different Laplacian realisations yield structurally different optimal partitions, exemplified by the ladder graph, which shows the significance of the chosen realization in the minimization problem.

This paper develops an existence theory for spectral minimal partitions on weighted combinatorial graphs, finite or countably infinite, extending the framework previously established for Euclidean domains and metric graphs. Given a partition $\mathcal P=(V_1,\dots,V_k)$ of a graph $G$ into connected, pairwise disjoint clusters, the authors study functionals of the form

$$\mathcal E_{k,p}^{\star}(\mathcal P)=\left(\frac{1}{k}\sum_{j=1}^k \lambda_1^{\star}(V_j)^p\right)^{1/p},\qquad p\in[1,\infty],$$

where $\lambda_1^{\star}(V_j)$ is the first positive eigenvalue of one of three self-adjoint realisations of the Laplacian on the induced subgraph: Dirichlet ($\lambda_1^{\mathrm D}$), Neumann ($\lambda_1^{\mathrm N}$), and boundaryless ($\lambda_1^{\mathrm B}$). The goal is to determine when the infimum of these energies over admissible partitions is actually attained.

Framework and the three Laplacian realisations

The graph $G$ carries vertex weights $\nu$ and edge weights $\omega$, with local finiteness $G$0. Edge weights induce a shortest-path metric via $G$1 as edge length, making $G$2 a metric measure space in the sense of Sturm (2608.19962). The energy form is $G$3, with form domain $G$4; boundary conditions at infinity (a Royden boundary) are encoded by choosing a closed intermediate form domain $G$5 between $G$6 and $G$7.

The Dirichlet energy is defined by restricting the ambient form to functions supported in $G$8: it decomposes as an internal Laplacian plus a Schrödinger potential $G$9 measuring the weighted degree flowing out across the boundary $$\mathcal E_{k,p}^{\star}(\mathcal P)=\left(\frac{1}{k}\sum_{j=1}^k \lambda_1^{\star}(V_j)^p\right)^{1/p},\qquad p\in[1,\infty],$$0. The boundaryless energy is the plain Laplacian of the induced subgraph $$\mathcal E_{k,p}^{\star}(\mathcal P)=\left(\frac{1}{k}\sum_{j=1}^k \lambda_1^{\star}(V_j)^p\right)^{1/p},\qquad p\in[1,\infty],$$1, ignoring the ambient graph; its first positive eigenvalue is well-defined whenever the subgraph is connected. The Neumann energy, following Shi–Yu, uses a non-isometric extension operator that averages values over the boundary $$\mathcal E_{k,p}^{\star}(\mathcal P)=\left(\frac{1}{k}\sum_{j=1}^k \lambda_1^{\star}(V_j)^p\right)^{1/p},\qquad p\in[1,\infty],$$2, so that Neumann conditions hold there; its associated quadratic form is shown to be closed.

A key structural result is the hierarchy: uniform Poincaré-type control (Assumption 3.4 for the boundaryless case) implies the analogous Neumann assumption (Assumption 3.6), which in turn implies canonical compactifiability (continuous embedding of $$\mathcal E_{k,p}^{\star}(\mathcal P)=\left(\frac{1}{k}\sum_{j=1}^k \lambda_1^{\star}(V_j)^p\right)^{1/p},\qquad p\in[1,\infty],$$3 into $$\mathcal E_{k,p}^{\star}(\mathcal P)=\left(\frac{1}{k}\sum_{j=1}^k \lambda_1^{\star}(V_j)^p\right)^{1/p},\qquad p\in[1,\infty],$$4), which yields compact embedding into $$\mathcal E_{k,p}^{\star}(\mathcal P)=\left(\frac{1}{k}\sum_{j=1}^k \lambda_1^{\star}(V_j)^p\right)^{1/p},\qquad p\in[1,\infty],$$5 and hence pure point spectrum for any admissible realisation. Geometrically, both Poincaré-type assumptions hold if $$\mathcal E_{k,p}^{\star}(\mathcal P)=\left(\frac{1}{k}\sum_{j=1}^k \lambda_1^{\star}(V_j)^p\right)^{1/p},\qquad p\in[1,\infty],$$6 has finite maximal non-contractible path length; sufficient conditions include finite total length $$\mathcal E_{k,p}^{\star}(\mathcal P)=\left(\frac{1}{k}\sum_{j=1}^k \lambda_1^{\star}(V_j)^p\right)^{1/p},\qquad p\in[1,\infty],$$7, or finite diameter together with finitely many independent cycles.

An explicit prism-graph example demonstrates that the three notions genuinely diverge: for certain induced subgraphs the ordering of cluster sizes by energy differs across all three functionals, foreshadowing that minimizers depend strongly on the chosen realization.

Existence theory

For finite graphs existence is trivial, since only finitely many partitions exist. The paper's contribution concerns infinite graphs, where the strategy is to reduce to the finite case via a topology of discrete convergence: a sequence of vertex sets converges iff their characteristic functions converge pointwise, a metrizable topology strictly weaker than Hausdorff convergence.

Three ingredients drive the argument. First, every sequence of $$\mathcal E_{k,p}^{\star}(\mathcal P)=\left(\frac{1}{k}\sum_{j=1}^k \lambda_1^{\star}(V_j)^p\right)^{1/p},\qquad p\in[1,\infty],$$8-partitions has a discretely convergent subsequence (a diagonal extraction over a countable vertex set), and limits of partitions whose limit clusters are nonempty are again partitions. Second, under the compactness assumptions above, normalized first-eigenvectors along minimizing sequences have subsequences converging in $$\mathcal E_{k,p}^{\star}(\mathcal P)=\left(\frac{1}{k}\sum_{j=1}^k \lambda_1^{\star}(V_j)^p\right)^{1/p},\qquad p\in[1,\infty],$$9, and Fatou's lemma gives lower semicontinuity of each energy, so $\lambda_1^{\star}(V_j)$0. Third, if between any two vertices there are only finitely many non-contractible paths — automatically true on ladders — then the class of connected-cluster partitions is sequentially closed under discrete convergence.

The resulting theorems state:

• Dirichlet case: if $\lambda_1^{\star}(V_j)$1 is compact, spectral minimal $\lambda_1^{\star}(V_j)$2-partitions exist for all $\lambda_1^{\star}(V_j)$3 and all $\lambda_1^{\star}(V_j)$4, including connected ones (disconnected clusters can be repaired without increasing the energy, using monotonicity of Dirichlet eigenvalues under inclusion).
• Boundaryless and Neumann cases: under the respective uniform Poincaré assumptions plus finite-noncontractible-paths, connected spectral minimal $\lambda_1^{\star}(V_j)$5-partitions exist for all $\lambda_1^{\star}(V_j)$6 and $\lambda_1^{\star}(V_j)$7.

Consequently, on any infinite graph with finite $\lambda_1^{\star}(V_j)$8, finite $\lambda_1^{\star}(V_j)$9, and finitely many non-contractible paths per pair of vertices, all three problems admit minimizers for every $\lambda_1^{\mathrm D}$0.

Spectral inequalities

The paper derives quantitative lower bounds on the optimal energies. Under finite $\lambda_1^{\mathrm D}$1 and finite total length,

$\lambda_1^{\mathrm D}$2

with the factor-$\lambda_1^{\mathrm D}$3 loss in the Dirichlet bound arising because the realizing paths for the inradii may share boundary edges. For unweighted finite graphs on $\lambda_1^{\mathrm D}$4 vertices, sharper bounds of the form $\lambda_1^{\mathrm D}$5 and $\lambda_1^{\mathrm D}$6 follow from Fiedler-type inequalities. The map $\lambda_1^{\mathrm D}$7 is continuous and non-decreasing.

A notable deviation from the domain and metric-graph theory: the inequality $\lambda_1^{\mathrm D}$8 familiar from Euclidean domains fails here; instead the paper proves the weaker bound $\lambda_1^{\mathrm D}$9, with the factor 2 forced by boundary edges shared between two clusters. A two-vertex graph shows even the uncorrected constant-1 version cannot hold: $\lambda_1^{\mathrm N}$0 while $\lambda_1^{\mathrm N}$1.

Ladder graphs: divergence of minimizers and failure of existence

For the finite ladder $\lambda_1^{\mathrm N}$2 with unit weights, exhaustive $\lambda_1^{\mathrm N}$3-partitions necessarily consist of clusters of the form $\lambda_1^{\mathrm N}$4 (a ladder of $\lambda_1^{\mathrm N}$5 rungs with a pendant path of length $\lambda_1^{\mathrm N}$6). A rearrangement argument based on the Perron–Frobenius positivity of the ground state shows that among such clusters with fixed vertex count, energy is minimized by concentrating rungs: $\lambda_1^{\mathrm N}$7 whenever $\lambda_1^{\mathrm N}$8. This yields a clean trichotomy of unique-or-characterized minimizers:

Energy	Minimizing 2-partition
Dirichlet	two copies of $\lambda_1^{\mathrm N}$9 ($\lambda_1^{\mathrm B}$0 even) or two copies of $\lambda_1^{\mathrm B}$1 ($\lambda_1^{\mathrm B}$2 odd)
Boundaryless	two path graphs $\lambda_1^{\mathrm B}$3, either horizontal or $\lambda_1^{\mathrm B}$4-shaped ($\lambda_1^{\mathrm B}$5)
Neumann	horizontal paths $\lambda_1^{\mathrm B}$6 only

Thus the three functionals select structurally different optimal partitions of the same graph — Dirichlet favors balanced ladder-like cells, boundaryless favors path-like cells in two configurations, and Neumann uniquely selects the horizontal cut, since the $\lambda_1^{\mathrm B}$7-shaped cell's Neumann eigenvalue strictly exceeds its boundaryless value by the equality-case analysis of Shi–Yu.

The infinite ladder $\lambda_1^{\mathrm B}$8 illustrates why the summability hypotheses are not technical artifacts. With unit weights, the boundaryless and Neumann energies are not even well defined on any infinite cluster: the variational infima equal zero but are not attained, since zero energy would require a non-$\lambda_1^{\mathrm B}$9 constant. In the Dirichlet case, if one admits infimum-of-spectrum as the energy, the global infimum is $G$0 for all $G$1, attained only for $G$2 (any bipartition into two semi-infinite ladders); for $G$3 no partition attains the infimum, mirroring a known phenomenon for unbounded metric graphs. Under finite $G$4 and $G$5, by contrast, all problems are well posed and minimizers exist for all $G$6.

The paper leaves open whether there exist graphs $G$7, integers $G$8, and exponents $G$9 for which the Neumann-optimal partition differs from both the Dirichlet and boundaryless optimizers — on ladders the Neumann minimizer coincides with one of the boundaryless minimizers.

Limitations and open questions

Several restrictions frame the results. Only exhaustive partitions into connected clusters are treated; relaxed notions (non-connected clusters, partitions of unity as in Osting–White–Oudet) are excluded, though Lemma 5.9 shows connectivity is not a genuine restriction for Dirichlet energies under compactness. Robin-type realisations are explicitly not considered. The existence theorems for Neumann and boundaryless energies require the uniform Poincaré assumptions, whose geometric sufficiency conditions (finite maximal path length, or finite diameter plus finitely many cycles) are restrictive; the authors note other routes to discreteness, such as sparsity conditions, but do not integrate them into the existence theory. The general claim that a discretely convergent minimizing sequence attaining a connected limit yields a minimizer is proven only under the finite-noncontractible-paths hypothesis ensuring closedness of the connected class. Whether the factor $\nu$0 in $\nu$1 is sharp, and whether the higher-order comparison $\nu$2 extends verbatim to the infinite setting beyond the first eigenvalue, are left unaddressed.

Conclusion

The paper supplies the first systematic existence theory for spectral minimal partitions of infinite weighted combinatorial graphs, built on canonical compactifiability, uniform Poincaré-type inequalities, discrete set convergence, and a closedness criterion for connected partitions expressed through non-contractible paths. It establishes quantitative lower bounds on all three optimal energies, identifies a structural correction (the factor 2) to the eigenvalue-partition inequality known from domains and quantum graphs, and demonstrates through the ladder model that the choice of Laplacian realisation qualitatively changes the optimizer — while showing precisely how, absent weight summability, the problem degenerates on infinite graphs.