Spectral Convergence of Graph Laplacians

• Spectral convergence of graph Laplacians is the study of how discrete eigenvalues and eigenfunctions approach their continuous counterparts via graph refinement and sampling techniques.
• Different graph constructions and normalization strategies, including ε-graphs, k-NN graphs, and density corrections, lead to various limiting operators such as the Laplace–Beltrami and fractal Laplacians.
• Quantitative convergence analyses employ kernel bandwidth, sample size, and error estimates to rigorously compare discrete spectral data with continuum limits in diverse geometric settings.

Searching arXiv for recent and foundational papers on spectral convergence of graph Laplacians to ground the article in the literature. Spectral convergence of graph Laplacians is the study of how spectral data of discrete graph operators—eigenvalues, eigenvectors, eigenspaces, resolvents, spectral projections, heat semigroups, or even empirical spectral distributions—approach the corresponding objects of a limiting operator under graph refinement, random sampling, bandwidth shrinkage, or structural degeneration. In the literature represented here, the limiting object may be the Laplace–Beltrami operator on a manifold, a weighted Laplacian under non-uniform sampling, a connection Laplacian on a vector bundle, a Laplacian on a fractal, an effective Laplacian on a coarsened graph, or another discrete graph Laplacian related by perturbation or contraction (Shi, 2015, Post et al., 2017, Singer et al., 2013, Koke, 26 Jan 2026, Bifulco et al., 2024).

1. Operator settings and model classes

The subject encompasses several distinct approximation regimes. One major regime begins with i.i.d. samples from a compact manifold and forms a weighted graph using an $\epsilon$-graph, a $k$-NN graph, a Gaussian kernel, or a symmetrized normalization. Another begins with an already discrete graph and varies its potential, self-loops, or intra-cluster weights. A third treats graphs as approximants of non-manifold spaces such as post-critically finite fractals. A fourth studies vector-valued analogues, where the scalar Laplacian is replaced by a connection Laplacian acting on sections of an associated vector bundle (Calder et al., 2019, Peoples et al., 2021, Post et al., 2017, Singer et al., 2013).

Setting	Discrete object	Limiting or comparison operator
Random samples on manifolds	$\epsilon$-graphs, $k$-NN graphs, Gaussian-kernel graph Laplacians	Laplace–Beltrami or weighted Laplacian (Calder et al., 2019, Cheng et al., 2021)
Manifolds with boundary	Symmetrized or truncated graph Laplacian	Neumann or Dirichlet Laplace–Beltrami operator (Peoples et al., 2021)
Vector-valued data on bundles	Connection graph Laplacian	Connection Laplacian on an associated bundle (Singer et al., 2013)
pcf fractals	Finite-dimensional weighted graph Laplacians	Fractal Laplacian via quasi-unitary equivalence (Post et al., 2017)
General discrete graphs	$L_0$ and $L_c$ on $\ell^2(X,m)$	Spectral comparison within the discrete category (Bifulco et al., 2024)
Tightly connected clusters	$L_\beta$ with diverging intra-cluster weights	Effective Laplacian on a coarsened graph (Koke, 26 Jan 2026)

This breadth is important because “spectral convergence of graph Laplacians” does not designate a single theorem. It designates a family of approximation theories whose hypotheses, normalizations, and limit operators differ substantially. In particular, the limit need not be the ordinary Laplace–Beltrami operator: non-uniform sampling produces weighted operators, bundle structure produces connection Laplacians, and purely discrete perturbations can yield exact comparison identities rather than continuum limits (Shi, 2015, Singer et al., 2013, Bifulco et al., 2024).

2. Modes of convergence and analytic mechanisms

The strongest results in this literature are not restricted to eigenvalue-by-eigenvalue convergence. For pcf fractals, the graph energy forms and fractal energy form are $\delta_m$-quasi-unitarily equivalent, with $\delta_m \to 0$ exponentially fast, yielding norm resolvent convergence,

$k$0

as well as operator-norm convergence of heat operators, spectral projections, and other functions of the Laplacian; eigenfunctions converge in energy norm (Post et al., 2017).

A second mechanism is resolvent convergence under graph contraction. When intra-cluster weights are scaled by a parameter $k$1, the corresponding Laplacians satisfy

$k$2

for $k$3, with strong resolvent convergence in the undirected setting and norm resolvent convergence for finite clusters; semigroup convergence follows as well (Koke, 26 Jan 2026).

A third mechanism is direct spectral comparison inside the discrete category. For general and possibly infinite discrete graphs with purely discrete spectrum, the perturbation identity

$k$4

holds, with the right-hand side equal to infinity if $k$5. The same work proves the discrete local Weyl law

$k$6

using Mercer’s theorem and the heat kernel diagonal, without Tauberian arguments or the continuous Weyl law (Bifulco et al., 2024).

In periodic metric graphs, convergence and comparison are organized through spectral band bracketing. Spectral bands of the metric Laplacian are localized by Dirichlet and Neumann eigenvalues on a finite fundamental domain, and the passage between discrete and metric spectra is mediated by the Cattaneo correspondence, which maps discrete band edges through $k$7 (Korotyaev et al., 2014).

These results show that spectral convergence is operator-theoretic before it is geometric. Depending on the setting, the decisive tool may be quasi-unitary equivalence, min–max comparison, RKHS methods, optimal transport, heat-kernel interpolation, quadratic-form monotonicity, or direct discrete heat-kernel expansions (Post et al., 2017, Peoples et al., 2021, Calder et al., 2019, Cheng et al., 2021, Koke, 26 Jan 2026, Bifulco et al., 2024).

3. Quantitative convergence from random samples

The best-developed quantitative theory concerns low-lying eigenpairs of graph Laplacians built from i.i.d. samples on a manifold. A common structure is the balance of bias from kernel smoothing against variance from empirical approximation, with graph bandwidth chosen as a function of sample size and intrinsic dimension.

Setting	Representative quantitative statement	Reference
Point Integral Method, non-uniform density	With probability at least $k$8, $k$9	(Shi, 2015)
Random geometric graphs	$\epsilon$0 for fixed low-lying spectrum	(Trillos et al., 2018)
$\epsilon$1-graphs and $\epsilon$2-NN graphs	Optimal rate, up to log factors, is $\epsilon$3 for both eigenvalues and eigenvectors	(Calder et al., 2019)
Gaussian kernelized graph Laplacian	If $\epsilon$4, eigenvalue rate is $\epsilon$5 and eigenvector rate is $\epsilon$6; if $\epsilon$7, both rates are $\epsilon$8	(Cheng et al., 2021)
$\epsilon$9 spectral convergence	Eigenvalue error $k$0 and eigenfunction sup-norm error $k$1, together with a heat-kernel reconstruction bound	(Dunson et al., 2019)

For the symmetric normalized graph Laplacian on a compact submanifold, one estimate is

$k$2

and when the data are sampled from a tubular neighborhood rather than the manifold itself, the required scaling changes from intrinsic dimension $k$3 to ambient dimension $k$4; the accompanying numerical study indicates the necessity of a denoising step before applying spectral algorithms (Wang, 2015).

The rates are not uniform across constructions. They depend on normalization, kernel regularity, bandwidth scaling, the choice between $k$5-graphs and $k$6-NN graphs, the target norm for eigenvector comparison, and whether the claim concerns pointwise, $k$7, $k$8, or $k$9 convergence. What is consistent across the literature is that low-lying spectral data converge under explicit sampling and bandwidth regimes, and that these regimes are controlled by intrinsic geometric dimension rather than ambient dimension in the noise-free setting (Shi, 2015, Trillos et al., 2018, Calder et al., 2019, Cheng et al., 2021, Dunson et al., 2019, Wang, 2015).

4. Density, normalization, boundary conditions, and cluster structure

Non-uniform sampling changes the continuum limit. In the Point Integral Method, as $L_0$0 and $L_0$1, the discrete spectrum converges to the weighted Neumann problem

$L_0$2

which reduces to the Laplace–Beltrami operator only when $L_0$3 is uniform (Shi, 2015). For Gaussian kernels, the density-corrected Laplacian is designed precisely so that, under non-uniform sampling, it recovers the Laplace–Beltrami operator rather than a Fokker–Planck-type limit (Cheng et al., 2021).

Boundary effects are also spectrally visible. On manifolds with boundary, the symmetrized graph Laplacian converges to the Laplace–Beltrami operator with homogeneous Neumann boundary conditions, while the Dirichlet problem is recovered by a truncated graph Laplacian obtained by retaining only points sufficiently far from the boundary. In that setting, the eigenvalue error is

$L_0$4

and with optimized $L_0$5 this becomes $L_0$6 (Peoples et al., 2021).

Normalization is not a cosmetic choice. In the three-parameter family

$L_0$7

the nearly separated two-cluster analysis shows that a uniform spectral gap above the second eigenvalue appears only in the balanced case $L_0$8; in unbalanced cases, only a ratio gap may remain (Hoffmann et al., 2019). Likewise, on unions of intersecting manifolds of different dimensions, the normalized graph Dirichlet energy converges to a dimension-adaptive limit on all manifolds simultaneously, whereas the unnormalized energy and its associated graph Laplacian asymptotically only see the variations within the manifold of the highest dimension (Bungert et al., 29 Sep 2025).

A plausible implication is that several recurrent identifications in applied work—normalized versus unnormalized, density-corrected versus standard, Neumann versus Dirichlet, balanced versus unbalanced cluster normalization—cannot be treated as minor implementation details. The limit operator, gap structure, and even which geometric component remains visible in the spectrum depend on them (Cheng et al., 2021, Peoples et al., 2021, Hoffmann et al., 2019, Bungert et al., 29 Sep 2025).

5. Extensions beyond smooth closed manifolds

The scope of spectral convergence is wider than smooth, boundaryless manifolds in Euclidean space. On pcf self-similar fractals with arbitrary Borel regular probability measure of full support, a sequence of finite-dimensional weighted graph Laplacians approximates the fractal Laplacian in norm resolvent sense, and therefore heat semigroups, spectral projections, Hausdorff convergence of spectra, and eigenfunctions in energy norm follow as consequences (Post et al., 2017).

For vector-valued problems, the principal-bundle framework extends the scalar theory to connection Laplacians on associated bundles. In this setting, eigenvalues and eigenvectors of discrete connection Laplacians converge, in the limit of infinitely many independent random samples, to the spectrum of the continuous connection Laplacian; the framework covers manifolds with and without boundary and non-uniform sampling, and subsumes Laplacian Eigenmaps, Diffusion Maps, Vector Diffusion Maps, and Orientable Diffusion Maps as special cases (Singer et al., 2013).

The 2025 Ricci-limit theory further weakens geometric regularity assumptions. Quantitative high-probability bounds are obtained for $L_0$9-neighborhood graph Laplacians on closed Riemannian manifolds under a uniform lower Ricci curvature bound, a positive lower volume bound, and an upper diameter bound, without injectivity radius or upper sectional curvature bounds. The same framework extends to non-collapsed Ricci limit spaces, yielding spectral approximation of weighted Laplacians on manifolds with non-smooth points (Inagaki, 9 Jun 2025).

Periodic metric graphs supply a different extension. There the spectrum consists of absolutely continuous bands plus flat bands, and each spectral band is localized by Dirichlet and Neumann eigenvalues on a finite fundamental domain; the discrete-metric relation is explicit through the Cattaneo correspondence (Korotyaev et al., 2014).

This collection suggests that spectral convergence is best understood as an approximation paradigm for Dirichlet forms and self-adjoint operators across manifold, fractal, metric-graph, and non-smooth metric-measure settings, rather than as a theorem tied to one ambient smooth category.

6. Discrete comparison principles, graph degeneration, and random-matrix limits

Not all spectral convergence problems pass through a continuum. In a general discrete-graph framework with countable vertex set $L_c$0, symmetric edge weights $L_c$1, vertex measure $L_c$2, and non-negative potential $L_c$3, the perturbation from $L_c$4 to $L_c$5 admits the exact comparison formula

$L_c$6

If $L_c$7, the eigenvalue differences form a summable null sequence, so $L_c$8 as $L_c$9; if all eigenvalues coincide, then $\ell^2(X,m)$0 must vanish, yielding an Ambarzumian-type rigidity statement (Bifulco et al., 2024).

Graph degeneration can also force effective reduction. When intra-cluster edge weights tend to infinity, the Laplacian converges in resolvent sense to the Laplacian of a graph in which each tightly connected cluster is collapsed to a single node. In the undirected case, only the cluster partition and aggregated inter-cluster weights enter. In the directed case, the limit depends on the left and right kernel structure of the cluster Laplacian blocks, so the effective graph is sensitive to asymmetry inside the highly connected subgraphs (Koke, 26 Jan 2026).

Self-loops alter the discrete spectrum in a different way. For graphs with self-loops, the Laplacian decomposes as

$\ell^2(X,m)$1

and the lifting construction produces a self-loop-free graph $\ell^2(X,m)$2 such that

$\ell^2(X,m)$3

If the graph is pseudo-connected, then the Laplacian is positive definite (Acikmese, 2015).

At the random-matrix scale, convergence may concern the empirical spectral distribution rather than individual low-lying eigenpairs. For centered scaled Laplacians of generalized Wigner matrices with a variance profile converging to a graphon, the empirical spectral distribution converges weakly in probability to a symmetric probability measure $\ell^2(X,m)$4, and the moments of $\ell^2(X,m)$5 admit graph-homomorphism formulas; spectral norm asymptotics are also derived (Chatterjee et al., 2020).

Across these discrete regimes, the limit object depends on what is being varied: perturbing a potential yields an exact eigenvalue-shift identity, strengthening clusters yields a coarse-grained Laplacian, adding self-loops yields a lifted-graph inclusion principle, and increasing matrix size with graphon-convergent variance profile yields convergence of spectral distributions. The common theme is not a single limiting geometry, but the controlled transfer of spectral information under graph modification, approximation, or degeneration.