---
title: Multiway Cheeger Constants
url: https://www.emergentmind.com/topics/multiway-cheeger-constants
type: topic
---

# Multiway Cheeger Constants

Multiway Cheeger constants are higher-order isoperimetric quantities that extend the classical Cheeger constant from a single cut to several disjoint pieces. In the standard graph-theoretic form they minimize the worst expansion among \(k\) disjoint sets; in signed graphs they minimize the worst signed bipartiteness ratio among \(k\) disjoint sub-bipartitions; on weighted manifolds they are \(k\)-way isoperimetric constants defined through Minkowski boundary measure; and on measurable domains they are max-type partition functionals built from the classical Cheeger constant of each piece [1401.3147] [1411.3530] [1307.3919] [1706.07282]. Across these settings, the common role of the theory is to encode multi-cluster structure and to relate it to higher spectral data, although the indexing conventions and admissible families differ substantially.

## 1. Foundational definitions and indexing conventions

A representative cross-section of definitions is shown below.

| Setting | Representative definition | Admissible family |
|---|---|---|
| Finite weighted graph | \(h(k)=\min_{S_1,\dots,S_k}\max_i \phi(S_i)\), with \(\phi(S)=|E(S,\overline S)|/\mathrm{vol}(S)\) | \(k\) non-empty pairwise disjoint subsets [1401.3147] |
| Signed graph | \(h_k^\sigma(\mu)=\min \max_i \beta^\sigma(V_{2i-1},V_{2i})\) | \(k\) pairwise disjoint sub-bipartitions [1411.3530] |
| Weighted manifold | \(h_k(M,\mu)=\inf \max_{0\le i\le k}\mu^+(A_i)/\mu(A_i)\) | \(k+1\) non-empty disjoint Borel subsets [1307.3919] |
| Measurable domain | \(h_k(\Omega)=\inf \max_i P(E_i)/|E_i|=\inf \max_i h_1(E_i)\) | \(k\) mutually disjoint measurable subsets [1706.07282] |

The graph-theoretic prototype is the higher-order Cheeger constant of Miclo and Lee–Oveis Gharan–Trevisan: for a weighted graph, one chooses \(k\) pairwise disjoint non-empty sets \(S_1,\dots,S_k\) and minimizes the largest conductance among them, with \(\phi(S)=|E(S,\overline S)|/\mathrm{vol}(S)\) [1401.3147]. Weighted forests use the same formal definition, written \(h_k(G)=\min_{(A_1,\dots,A_k)\in\mathcal P_k(V)}\max_i \phi(A_i)\) with \(\phi(A)=w(\partial A)/\mu(A)\) [2510.06301].

The indexing is not uniform across the literature. On weighted manifolds, Funano defines the \(k\)-way isoperimetric constant using \(k+1\) disjoint sets, so \(h_1(M,\mu)\) is already the 2-way Cheeger constant [1307.3919]. On Euclidean domains, the higher Cheeger constant \(h_k(\Omega)\) is the infimum of the maximum of the classical Cheeger constants of \(k\) mutually disjoint subsets [1706.07282]. On signed graphs, the natural objects are not subsets but sub-bipartitions \((V_{2i-1},V_{2i})\), because balance rather than mere connectivity is the governing structural notion [1411.3530].

Despite these differences, a shared structural feature is monotonicity: the sequence is nondecreasing in the primal cases \(h_1\le h_2\le\cdots\), while dual multiway Cheeger constants decrease with \(k\) [1401.3147] [1307.3919] [1411.3530].

## 2. Spectral and variational correspondences

The central reason multiway Cheeger constants matter is that they encode higher eigenmodes rather than only the first nontrivial one. For finite weighted graphs, Liu recalls the higher-order Cheeger inequality
\[
\frac{\lambda_k}{2}\le h(k)\le C k^2\sqrt{\lambda_k},
\]
where \(\lambda_k\) is the \(k\)-th eigenvalue of the normalized Laplacian and \(C\) is universal [1401.3147]. This is the direct higher-order analogue of the classical Cheeger inequality.

On signed graphs, Atay and Liu obtain the parallel bound
\[
\frac{\lambda_k(\Delta^\sigma)}{2}\le h_k^\sigma(\mu_d)\le C k^3\sqrt{\lambda_k(\Delta^\sigma)},
\]
and likewise for the non-normalized signed Laplacian [1411.3530]. The quantity \(h_k^\sigma\) vanishes exactly when the signed graph has at least \(k\) balanced connected components, so the spectral information is tied to structural balance rather than ordinary disconnection [1411.3530].

On weighted manifolds with nonnegative Bakry–Émery Ricci curvature, Funano proves a genuine two-sided comparison:
\[
(80k^3)^{-1}\sqrt{\lambda_k(M,\mu)}\le h_k(M,\mu)\le c\,k^3\sqrt{\lambda_k(M,\mu)},
\]
and also
\[
\lambda_k(M,\mu)\le \exp(c k)\,\lambda_1(M,\mu),\qquad
h_k(M,\mu)\le k^3\exp(c k)\,h_1(M,\mu).
\]
Thus \(h_k(M,\mu)\) and \(\sqrt{\lambda_k(M,\mu)}\) are equivalent up to polynomial factors in \(k\) under \(\mathrm{Ric}_\mu\ge 0\) [1307.3919].

The variational side becomes especially rigid on forests. For weighted forests, the 2025 minimax result gives
\[
h_k(G)=\ell_k(G)=\underline{\ell_k}(G),
\]
a combinatorial analogue of the Courant–Fischer–Weyl minimax principle, and further
\[
h_k(G)=\lambda_k^{\mathrm{ind}}(\Delta_1)=\underline{\lambda_k^{\mathrm{ind}}}(\Delta_1)
\]
for any admissible index [2510.06301]. On measurable domains, the analogous spectral partition object is
\[
\mathfrak{L}_k(p;\Omega):=\inf_{(E_1,\dots,E_k)}\max_i \lambda_1(p;E_i),
\]
and the higher Cheeger constant satisfies
\[
\lim_{p\to 1}\mathfrak{L}_k(p;\Omega)=h_k(\Omega),
\]
so the \(p\to 1\) limit of spectral minimal \(k\)-partitions is exactly the multiway Cheeger problem [1706.07282].

These results collectively indicate that multiway Cheeger constants are not merely combinatorial cut parameters. They are higher-order isoperimetric invariants with direct min–max, Rayleigh-quotient, and spectral-partition interpretations.

## 3. Signed, dual, and projective-space extensions

The unsigned higher-order theory has two distinct directions: the primal side, controlled by small Laplacian eigenvalues and sparse cuts, and the dual side, controlled by large Laplacian eigenvalues and bipartite structure. Liu formalizes the latter via multi-way dual Cheeger constants
\[
\overline{h}(k)=
\max_{\{(V_{2i-1},V_{2i})\}\in \mathrm{Pair}(k)}
\min_i \overline{\phi}(V_{2i-1},V_{2i}),
\qquad
\overline{\phi}(V_1,V_2)=\frac{2|E(V_1,V_2)|}{\mathrm{vol}(V_1\cup V_2)},
\]
which quantify how well the graph contains \(k\) disjoint bipartite-like regions [1401.3147]. The associated higher-order dual Cheeger inequality is
\[
\frac{2-\lambda_{N-k+1}}{2}\le 1-\overline{h}(k)\le C k^3\sqrt{2-\lambda_{N-k+1}}.
\]
Moreover,
\[
\overline{h}(k)\le 1-h(k),\qquad
\overline{h}(k)\ge \frac12(1-h(k)),
\]
and for bipartite graphs one has the exact duality
\[
h(k)+\overline{h}(k)=1.
\]
Thus the dual constants are not auxiliary; they are the upper-spectrum counterpart of the ordinary multiway Cheeger constants [1401.3147].

Signed graphs absorb both the primal and dual viewpoints into a single switching-invariant framework. For a signed graph \(\Gamma=(G,\sigma)\), Atay and Liu define
\[
h_k^\sigma(\mu):=
\min_{\{(V_{2i-1},V_{2i})\}_{i=1}^k}
\max_i \beta^\sigma(V_{2i-1},V_{2i}),
\]
where \(\beta^\sigma\) is a signed bipartiteness ratio that penalizes positive edges across the two sides, negative edges within each side, and boundary edges leaving the cluster [1411.3530]. They prove that
\[
h_k^\sigma(\mu)=0
\quad\Longleftrightarrow\quad
\Gamma\text{ has at least }k\text{ balanced connected components},
\]
and that \(h_k^\sigma\) is switching invariant [1411.3530].

Equivalent formulations show that signed multiway Cheeger constants interpolate between ordinary boundary expansion and internal frustration. In particular,
\[
h_1^\sigma(\mu)=\min_{\emptyset\neq S\subseteq V}\alpha^\sigma(S),
\qquad
\alpha^\sigma(S)=\frac{2e_{\min}(S)+|E(S,\overline S)|}{\mathrm{vol}_\mu(S)},
\]
where \(e_{\min}(S)\) is the frustration index of the induced subgraph on \(S\) [1411.3530]. This unifies classical Cheeger constants, bipartiteness measures, and structural balance.

A notable geometric consequence of the dual and signed theories is that the correct clustering metric is no longer the ordinary spherical metric. Liu’s dual theory and Atay–Liu’s signed theory both use a metric induced from real projective space, so antipodal directions are identified. This is natural because bipartite or balanced structure is invariant under sign reversal inside a cluster [1401.3147] [1411.3530].

## 4. Continuum formulations on manifolds and domains

In the manifold setting, the multiway Cheeger constant becomes a \(k\)-way isoperimetric constant defined with Minkowski boundary measure:
\[
h_k(M,\mu):=
\inf \max_{0\le i\le k}\frac{\mu^+(A_i)}{\mu(A_i)},
\]
where the infimum ranges over \(k+1\) non-empty disjoint Borel subsets of a closed weighted manifold \((M,\mu)\) [1307.3919]. Funano studies this under the curvature condition \(\mathrm{Ric}_\mu\ge 0\), equivalently \(CD(0,\infty)\), and obtains dimension-free control of higher eigenvalues and higher isoperimetric constants. The resulting picture is that multiway separation, concentration, higher spectrum, and higher isoperimetry are quantitatively equivalent up to factors depending only on \(k\) [1307.3919].

For bounded measurable domains \(\Omega\subset\mathbb R^N\), the higher Cheeger constants are defined by
\[
h_k(\Omega)=
\inf\left\{
\max_{i=1,\dots,k}\frac{P(E_i)}{|E_i|}:
E_i\subset\Omega,\ |E_i|>0,\ E_i\cap E_j=\emptyset\ (i\neq j)
\right\},
\]
or equivalently by replacing \(P(E_i)/|E_i|\) with \(h_1(E_i)\) [1706.07282]. Here the geometry of minimizers is subtler than in the classical \(k=1\) problem because the objective depends only on the largest ratio among the pieces.

To control this nonuniqueness, the theory introduces \(n\)-adjusted Cheeger \(k\)-tuples. A 1-adjusted tuple requires each \(E_i\) to be a Cheeger set of the leftover region \(\Omega\setminus \bigcup_{j\neq i}E_j\); higher adjustment requires every subcollection to solve the appropriate lower-order problem in the corresponding leftover domain [1706.07282]. For every bounded measurable \(\Omega\), every \(k\), and every \(n\in\{1,\dots,k\}\), there exists an \(n\)-adjusted Cheeger \(k\)-tuple [1706.07282].

When \(\Omega\) is bounded and open, 1-adjusted tuples enjoy regularity: each component is \((\Lambda,r_0)\)-perimeter minimizing with \(\Lambda=h_k(\Omega)\) and \(r_0=1/h_k(\Omega)\); \(\partial^*E_i\cap\Omega\) is \(C^{1,\gamma}\) for every \(\gamma\in(0,1/2)\); and the singular set has Hausdorff dimension at most \(N-8\) [1706.07282]. In the planar case, free boundaries are arcs of circles with curvature \(h_1(E_i)\), and contact interfaces between different components have constant curvature in 2-adjusted tuples [1706.07282].

The continuum spectral correspondence is formulated through spectral minimal \(k\)-partitions rather than through the raw \(k\)-th eigenvalue of the \(p\)-Laplacian. With
\[
\mathfrak{L}_k(p;\Omega)=
\inf_{(E_1,\dots,E_k)}
\max_i \lambda_1(p;E_i),
\]
one has
\[
\lim_{p\to1}\mathfrak{L}_k(p;\Omega)=h_k(\Omega),
\]
which is the direct \(k\)-part counterpart of the classical Cheeger–\(p\)-Laplacian relation [1706.07282].

## 5. Refinements on forests, sparse graphs, and low-cycle regimes

The higher-order theory becomes especially explicit on acyclic or nearly acyclic graphs. For weighted forests, the 2025 minimax theorem establishes
\[
h_k(G)=\ell_k(G)=\underline{\ell_k}(G),
\]
where \(\ell_k\) is a max–min Cheeger functional over \((n-k+1)\)-subpartitions and \(\underline{\ell_k}\) is the corresponding Dirichlet Cheeger \(k\)-constant [2510.06301]. This is presented as the first combinatorial analogue of the Courant–Fischer–Weyl minimax principle. In the same setting, the forest 1-Laplacian variational eigenvalues are independent of the choice of typical indexes, and
\[
\lambda_k^\gamma(\Delta_1)
=
\lambda_k^{\gamma^+}(\Delta_1)
=
\lambda_k^{\text{Y-ind}}(\Delta_1)
=
h_k(G)
\]
for forests [2510.06301].

The forest case is also the anchor point for refined \(p\)-Laplacian bounds. Assuming \(\mu_v=\sum_{u\sim v}w_{uv}\), the paper proves
\[
\frac{2^{p-1}}{p^p}h_k(G)^p\le \lambda_k(\Delta_p)\le 2^{p-1}h_k(G)
\]
for every weighted forest and every \(p\ge1\) [2510.06301]. For general graphs, the same paper introduces the cycle rank \(\beta\) and shows
\[
h_{k-\beta}(G)\le \lambda_k(\Delta_1)\le h_k(G),
\]
together with
\[
\frac{2^{p-1}}{p^p}h_{k-\beta}(G)^p\le \lambda_k(\Delta_p)\le 2^{p-1}h_k(G)
\]
for all \(p\ge1\) [2510.06301]. This suggests that loop structure controls how far the exact forest theory can be extended.

A complementary refinement uses the cyclomatic number \(\ell=|E|-|V|+1\). Ge proves
\[
p_{k-\ell}(G)\le \sqrt{2T_G\,\lambda_k(G)},
\]
where \(p_k(G)\) is the \(k\)-way Cheeger constant in that paper’s notation and \(T_G=\max_i d(i)/p_i\) [2409.07097]. In the normalized case this removes the usual \(k\)-dependent constant from higher-order Cheeger upper bounds at the price of shifting the index by \(\ell\). The same work also gives a lower bound
\[
p_k(G)\ge (T_{\min}-\theta)(1-\gamma),
\]
and in the normalized setup
\[
p_k(G)\ge (1-\gamma)\min\{\lambda_2(G),\,2-\lambda_n(G)\},
\]
linking all multiway Cheeger constants to the spectral radius of the normalized adjacency matrix [2409.07097].

Not every Cheeger-type result near the top of the spectrum is genuinely multiway. In finite Cayley graphs, Biswas derives an explicit bound away from \(-1\) using the ordinary vertex Cheeger constant and the group-theoretic characterization of bipartiteness, but explicitly notes that the paper does not develop or use multiway Cheeger constants in the Cayley setting [1803.03969]. This provides a useful boundary line: one-way and dual-edge phenomena can motivate the multiway theory without constituting it.

## 6. Algorithms, projective clustering, and higher-dimensional analogues

The theory is not only structural; it also supports concrete partitioning procedures. In the signed and dual settings, the algorithmic core is spectral embedding into \(\mathbb R^k\), normalization onto the sphere, and clustering with a projective metric
\[
d_\Phi(u,v)=
\min\left\{
\left\|\frac{\Phi(u)}{\|\Phi(u)\|}-\frac{\Phi(v)}{\|\Phi(v)\|}\right\|,
\left\|\frac{\Phi(u)}{\|\Phi(u)\|}+\frac{\Phi(v)}{\|\Phi(v)\|}\right\|
\right\},
\]
which is the distance induced from \(\mathbb{RP}^{k-1}\) [1411.3530]. This is used to produce \(k\) almost-balanced or bipartite-like subgraphs with guarantees matching the higher-order Cheeger bounds [1411.3530] [1401.3147].

A different algorithmic formulation appears in multiway spectral graph partitioning. Damle, Minden, and Ying use the top \(k\) eigenvectors of the normalized adjacency matrix and encode the partition in an indicator matrix \(\Psi\) obtained by approximating the eigenvector matrix by \(\Psi Q\) with \(Q\in O(k)\) [2302.03615]. They define two cut functions,
\[
\Psi\mathrm{Cut}_G,\qquad \Phi\mathrm{Cut}_G,
\]
and a spectral distance to \(k\)-partitionability,
\[
L_k=\left(\sum_{\nu=2}^k(1-\lambda_\nu)^2\right)^{1/2},
\]
with
\[
L_k\le \Psi\mathrm{Cut}_G,\qquad L_k\le \Phi\mathrm{Cut}_G.
\]
The associated alternating semi-sparse orthogonal approximation algorithm is presented as a simple spectral method for multiway partitioning [2302.03615].

The broader literature also distinguishes multiway from higher-dimensional generalizations. For \(k\)-dimensional simplicial complexes, Gundert and Szedlák define a combinatorial Cheeger constant \(h(X)\) through partitions of the vertex set into \(k+1\) blocks and counts of rainbow \(k\)-simplices, and prove
\[
\lambda(X)\le h(X)
\]
for arbitrary complexes, together with the sharper bound
\[
\lambda(X)\le h'(X)
\]
for a refined cochain-based parameter \(h'(X)\) [1401.2290]. This is a different axis of generalization: rather than multiple disjoint subsets in a graph, the expansion object is a higher-dimensional face structure. The literature therefore separates three themes that are sometimes conflated: higher-order graph Cheeger constants, dual or signed multiway Cheeger constants, and higher-dimensional Cheeger constants on simplicial complexes [1401.2290] [1411.3530] [1401.3147].

In that sense, “multiway Cheeger constants” names a family rather than a single invariant. What persists across the family is the passage from one cut to many, the replacement of first-eigenvalue geometry by higher-order spectral structure, and the use of isoperimetric quantities to detect multiple clusters, multiple balanced components, or multiple near-bipartite regions.

Source: https://www.emergentmind.com/topics/multiway-cheeger-constants