Combinatorial Laplacians for Complex Pairs

• Combinatorial Laplacians for complex pairs are discrete Hodge-type operators defined on chain groups of pairs, generalizing classical Laplacians to capture relative homology.
• They extend absolute simplicial Laplacians by incorporating relative and persistent formulations that analyze spectral invariants, stability, and topological features.
• Recent advances embed these operators in a Hilbert complex framework, clarifying spectral gaps, stability conditions, and providing matrix-tree theorem generalizations.

Searching arXiv for the most relevant papers on combinatorial Laplacians for complex pairs and related generalized/relative formulations. arXiv search: (Zhan et al., 22 Jul 2025) combinatorial Laplacians relative homology complex pairs. Combinatorial Laplacians for complex pairs are discrete Hodge-type operators attached not only to a single simplicial or cellular complex, but also to data such as a simplicial pair $(X,A)$, an inclusion $K\subset L$, a primal–dual pair $(K,\star K)$, or a complex with a distinguished boundary subcomplex. In the relative simplicial setting, the operator acts on relative chain groups $C_k(X,A)$ and detects relative homology $H_k(X,A)$; in persistent settings, the operator is instead built on $C_k(K)$ and uses ambient $(k+1)$-chains in $L$ whose boundary returns to $K$. Recent work places these constructions in a broader Hilbert-complex framework and clarifies which spectral features are topological, which are boundary-condition dependent, and which fail to be monotone or stable for pairs (Zhan et al., 22 Jul 2025, Wolf et al., 24 Sep 2025).

1. Absolute simplicial Laplacians as the base model

For a finite simplicial complex $K$, the standard chain groups are

$K\subset L$0

with boundary maps

$K\subset L$1

when $K\subset L$2 is obtained by deleting the $K\subset L$3-th vertex of $K\subset L$4. After choosing inner products on chain groups, one obtains adjoints $K\subset L$5, and the combinatorial Hodge Laplacian is

$K\subset L$6

Its kernel recovers simplicial homology: $K\subset L$7 This is the discrete model that later pair constructions generalize (Wolf et al., 24 Sep 2025).

The same structure can be written on cochains. In the weighted simplicial framework, one introduces coboundaries $K\subset L$8, adjoints $K\subset L$9, and the up-, down-, and full Laplacians

$(K,\star K)$0

These operators are self-adjoint and non-negative, and their kernel is identified with reduced cohomology by the discrete Hodge theorem (Horak et al., 2011).

This absolute theory already contains the algebraic pattern that persists for pairs: an upper term coming from adjacent higher-dimensional cells, a lower term coming from adjacent lower-dimensional cells, and a harmonic kernel representing a topological quotient.

2. Relative Laplacians on simplicial complex pairs

For a simplicial pair $(K,\star K)$1, with $(K,\star K)$2 a subcomplex, the basic object is the relative chain group

$(K,\star K)$3

The paper "Combinatorial Laplacians and relative Homology of complex pairs" (Zhan et al., 22 Jul 2025) identifies this canonically with the free $(K,\star K)$4-module on simplices of $(K,\star K)$5 in degree $(K,\star K)$6: $(K,\star K)$7 Under this identification, the relative boundary map is

$(K,\star K)$8

Relative boundaries are therefore obtained from ordinary simplicial boundaries by discarding those faces that lie in $(K,\star K)$9.

Over $C_k(X,A)$0, the basis $C_k(X,A)$1 is declared orthonormal. The adjoint of the relative boundary is the relative coboundary

$C_k(X,A)$2

The $C_k(X,A)$3-dimensional relative Laplacian is then

$C_k(X,A)$4

Its two summands are the up-down and down-up Laplacians,

$C_k(X,A)$5

The entrywise formulas make the effect of the subcomplex $C_k(X,A)$6 explicit. For $C_k(X,A)$7,

$C_k(X,A)$8

and

$C_k(X,A)$9

The diagonal term of the lower part is reduced from $H_k(X,A)$0 to the number of codimension-one faces not in $H_k(X,A)$1, and off-diagonal terms depend on whether the common $H_k(X,A)$2-face lies inside or outside $H_k(X,A)$3. In the graph case, $H_k(X,A)$4 is exactly a Dirichlet Laplacian on vertices not in $H_k(X,A)$5 (Zhan et al., 22 Jul 2025).

3. Relative Hodge theory, spectral gaps, and matrix-tree theorems

The central topological statement for relative simplicial Laplacians is the relative Hodge theorem: $H_k(X,A)$6 Equivalently,

$H_k(X,A)$7

Thus the smallest eigenvalue

$H_k(X,A)$8

satisfies

$H_k(X,A)$9

This makes relative Laplacians a spectral model for relative topology, exactly as ordinary combinatorial Laplacians model ordinary homology (Zhan et al., 22 Jul 2025).

The same work develops a relative matrix-tree theorem. For a pair $C_k(K)$0, one defines $C_k(K)$1-dimensional relative spanning forests and relative spanning trees by homological conditions on subcomplexes $C_k(K)$2 with $C_k(K)$3. If $C_k(K)$4, then the product of all nonzero eigenvalues of $C_k(K)$5 is expressed as a weighted sum over relative spanning trees and relative spanning forests: $C_k(K)$6 This generalizes both the Duval–Klivans–Martin higher-dimensional matrix-tree theorem and Chung’s result for graph pairs (Zhan et al., 22 Jul 2025).

The same paper proves lower bounds for the relative spectral gap. If $C_k(K)$7 is a $C_k(K)$8-th discrete boundary, then

$C_k(K)$9

where $(k+1)$0 is the associated subcomplex, $(k+1)$1 is the maximum dimension of a missing face of $(k+1)$2, and $(k+1)$3. For flag complex pairs, one obtains

$(k+1)$4

These bounds yield sufficient conditions for $(k+1)$5. A further comparison theorem relates $(k+1)$6 to the ordinary gap $(k+1)$7, showing that sufficiently large ambient spectral gap forces vanishing of relative homology (Zhan et al., 22 Jul 2025).

4. Persistent Laplacians: pair operators that are not relative Laplacians

A distinct construction arises for inclusions $(k+1)$8. The persistent combinatorial Laplacian is not the Laplacian of the relative complex $(k+1)$9. Instead, it is built on $L$0, but uses ambient $L$1-chains in $L$2 whose boundary lands in $L$3: $L$4 The persistent Laplacian is

$L$5

Its kernel gives image-persistent homology

$L$6

in finite dimensions, and more generally under a closed-range hypothesis (Wolf et al., 24 Sep 2025).

The paper "Persistent Laplacians: properties, algorithms and implications" (Mémoli et al., 2020) proves the persistent analogue of the Hodge theorem: $L$7 and shows that the persistent up-Laplacian is a Schur complement of the ordinary up-Laplacian on $L$8. This gives an explicit linear-algebraic interpretation of the pair operator as elimination of simplices outside $L$9.

The later generalization "Generalized Persistent Laplacians and their Spectral Properties" (Wolf et al., 24 Sep 2025) embeds this construction into a Hilbert-complex framework. For an inclusion $K$0 of Hilbert complexes, the auxiliary space is

$K$1

and the persistent Laplacians are defined by quadratic forms with formal expressions

$K$2

A central clarification is that the full persistent Laplacian is generally not monotone or stable, even for finite simplicial complexes. By contrast, the up- and down-persistent Laplacians are monotone under nested inclusions and stable under interleaving distance. Moreover, their nonzero spectra determine the nonzero spectrum of the full operator. This corrects a common misconception: the spectrally well-behaved objects for inclusions are the separated up and down parts, not the full persistent Laplacian as a standalone invariant (Wolf et al., 24 Sep 2025).

5. Operator-theoretic generalizations and infinite-dimensional settings

The generalized theory replaces finite chain groups by Hilbert spaces and ordinary differentials by densely defined closed operators. A Hilbert complex is

$K$3

where each $K$4 is a Hilbert space and each

$K$5

is densely defined and closed, with

$K$6

This setup allows self-adjoint Laplacians to be defined through quadratic forms even when the naïve operator sum

$K$7

is problematic on domain intersections (Wolf et al., 24 Sep 2025).

For a Hilbert complex, the up-, down-, and full Laplacians are induced by

$K$8

and

$K$9

The kernel satisfies

$K$0

and the generalized Hodge theorem gives

$K$1

When images are closed, this reduces to ordinary homology (Wolf et al., 24 Sep 2025).

A related operator-theoretic development studies Hodge Laplacians on general countable weighted simplicial complexes and distinguishes Dirichlet and Neumann realizations via quadratic forms. It defines

$K$2

with corresponding self-adjoint operators $K$3 and $K$4, $K$5. The paper does not explicitly formulate a theory of pairs $K$6, but it treats Dirichlet and Neumann domains as discrete analogues of $K$7 and $K$8. This suggests an operator-theoretic route to pair Laplacians through closed form domains rather than purely finite-dimensional matrix formulas (Bartmann et al., 11 Aug 2025).

6. Other pair formalisms: primal–dual pairs, boundary-value pairs, and cellular pairs

A different notion of complex pair appears in discrete exterior calculus on a primal complex $K$9 together with its combinatorial dual $K\subset L$00. In that setting, the Laplacian depends intrinsically on both complexes because $K\subset L$01 is defined from incidence and $K\subset L$02 from the Hodge star associated with the primal–dual correspondence. For dual scalar fields $K\subset L$03, the operator is

$K\subset L$04

a weighted nearest-neighbor Laplacian on the dual $K\subset L$05-skeleton. The construction is therefore a Laplacian for the pair $K\subset L$06, not for a single incidence structure alone (Calcagni et al., 2012).

For complexes with boundary, a boundary-value formulation uses a pair $K\subset L$07 where $K\subset L$08 is a boundary subcomplex. In that setting, one considers the relative cochain complex

$K\subset L$09

and the corresponding Laplacian $K\subset L$10. In degree zero this is a discrete Dirichlet Laplacian. The same formalism yields a combinatorial Dirichlet-to-Neumann operator from the quadratic form of the discrete Dirichlet problem and leads to a gluing formula for determinants of combinatorial Laplacians under boundary identification (Reshetikhin et al., 2014).

In higher-dimensional cell complexes, the pair viewpoint appears through relative interpretations of Laplacian minors. For a finite CW complex, the combinatorial Laplacian on $K\subset L$11-chains is

$K\subset L$12

The higher-dimensional Kirchhoff-type formulas express coefficients of its characteristic polynomial by sums over forests and spanning coforests, and the relevant minors are identified with boundary maps of pairs

$K\subset L$13

so that

$K\subset L$14

The paper does not formulate a full standalone theory of relative Laplacians for $K\subset L$15, but its matrix-tree formulas are already organized around pair homology groups (Cappell et al., 2015).

7. Conceptual distinctions and common confusions

Several non-equivalent constructions are called “Laplacians for pairs,” and the distinctions are substantive rather than terminological.

Construction	State space	Topological or spectral target
Relative simplicial Laplacian	$K\subset L$16	$K\subset L$17
Persistent Laplacian	$K\subset L$18 with ambient $K\subset L$19-chains	$K\subset L$20
Primal–dual DEC Laplacian	$K\subset L$21 cochains/forms	Hodge operator on a primal–dual pair
Dirichlet pair Laplacian	$K\subset L$22	Boundary-value problem and DN operator

The first misconception is to identify persistent Laplacians with relative Laplacians. The generalized persistent theory states explicitly that the persistent Laplacian for $K\subset L$23 is not a Laplacian on the quotient complex $K\subset L$24, nor simply the relative complex $K\subset L$25; it is a Laplacian on $K\subset L$26 whose upper part is induced from ambient chains in $K\subset L$27 whose boundary returns to $K\subset L$28 (Wolf et al., 24 Sep 2025).

The second misconception is to treat all pair Laplacians as equally robust spectral invariants. Relative Laplacians on $K\subset L$29 inherit the ordinary Hodge paradigm directly, while full persistent Laplacians fail monotonicity and general stability. In the persistent setting, the robust objects are the up- and down-persistent pieces, whose spectra nevertheless determine the nonzero spectrum of the full operator (Wolf et al., 24 Sep 2025).

The third misconception is that pair Laplacians are necessarily quotient-based. The primal–dual DEC Laplacian is pair-based in a different sense: it uses two coupled complexes $K\subset L$30 and $K\subset L$31. Boundary-value and gluing formalisms likewise use a pair $K\subset L$32, but the operator is organized by vanishing conditions, normal components, and Dirichlet-to-Neumann maps rather than by relative chain quotients (Calcagni et al., 2012, Reshetikhin et al., 2014).

Taken together, these constructions show that “combinatorial Laplacians for complex pairs” is not a single theory but a family of related Hodge-type frameworks. The relative simplicial Laplacian of $K\subset L$33 is the direct discrete model for relative homology; persistent Laplacians encode image persistence for inclusions; primal–dual Laplacians couple a complex to its dual; and Dirichlet pair Laplacians encode boundary-value problems and gluing. The recent literature makes these differences explicit and, in the relative and generalized persistent settings, connects them to precise spectral, homological, and enumerative statements (Zhan et al., 22 Jul 2025, Wolf et al., 24 Sep 2025).