What is a proper graph Laplacian? An operator-theoretic framework for graph diffusion

Lemma 1

(Monotonicity under edge addition) [19,39] If G ′ = ( V , E ′ ) is obtained from G = ( V , E ) by adding edges ( E ⊆ E ′ ), then L ( G ′ ) − L ( G ) is positive semidefinite. Consequently, μ k ( G ′ ) ≥ μ k ( G ) for all k , and in particular μ n ( G ′ ) ≥ μ n ( G ) .

Proof

Each added edge { i , j } contributes the rank-1 PSD Laplacian L { i , j } = ( e i − e j ) ( e i − e j ) ⊤ . Summing such contributions yields L ( G ′ ) − L ( G ) ≽ 0 . The eigenvalue monotonicity follows from the Courant-Fischer characterization.□

Lemma 2

(Upper bound by maximum degree) [19,82] For any graph G , μ n ( G ) ≤ 2 Δ ( G ) , where Δ ( G ) is the maximum degree.

Proof

By Gershgorin disks for the Laplacian, every eigenvalue lies in [ 0 , 2 Δ ] . Equivalently, x ⊤ L x = ∑ { i , j } ∈ E ( x i − x j ) 2 ≤ 2 Δ ∑ i x i 2 .□

Lemma 3

(Cheeger upper bound) [17,19] Let ϕ ( G ) ≔ min ∅ ≠ S ⊊ V ∣ E ( S , S ¯ ) ∣ min { vol ( S ) , vol ( S ¯ ) } , where vol ( S ) = ∑ i ∈ S k i . Then, the (combinatorial) Laplacian satisfies μ 2 ( G ) ≤ 2 ϕ ( G ) .

Proof

This classical bound follows by taking the Fiedler Rayleigh quotient with a test function f that is approximately constant on S and on S ¯ with opposite signs, and optimizing over S (refer, e.g., standard Cheeger-type inequalities for the combinatorial Laplacian [4,19,82]).□

Definition 4

(Graph Laplacian). A matrix L ∈ C n × n associated with a graph G = ( V , E ) is said to be a Laplacian operator on G if there exist two linear operators

acting between the space of complex-valued functions on vertices and that on a family of subgraph structures Σ of G (for instance, edges, paths, cycles, or higher-order motifs), such that

Here Σ is a family of graph elements that connect groups of vertices, such as edges, paths, cycles, or any other subgraph structures of G . The gradient operator grad maps vertex functions into functions defined on Σ , thus quantifying the local variations in the field between connected graph elements, while div aggregates those variations back onto the vertices.

Remark 5

The structural definition above differs fundamentally from the usual algebraic characterizations of a Laplacian, such as those of Balaji-Bapat [8] and Perez Riascos et al. [79]/Devriendt [27]. Because L = div grad and grad ( 1 ) = 0 , every proper Laplacian automatically satisfies L 1 = 0 , i.e., it has zero row-sum, and its nullspace always contains the constants, so that rank ( L ) ≤ n − 1 for a connected graph. No other matrix property of Balaji and Bapat [8] and Perez Riascos et al. [79]/Devriendt [27] is required. In particular, positive semidefiniteness, symmetry, and nonpositive off-diagonal entries are not implied by the operator form div grad . They arise only when one further demands div = grad * with respect to some inner product, in which case L = grad * grad becomes self-adjoint and positive semidefinite in that metric. Without this additional assumption, L may be nonsymmetric or even indefinite, as exemplified by the degree-biased Laplacians or other anisotropic diffusions. Finally, the requirement that grad acts on a family Σ of subgraph structures ensures that L is not an arbitrary zero-row-sum matrix, but one generated by local or higher-order interactions prescribed by the graph G .

Definition 6

(Graph gradient). The operator grad is a proper gradient if it satisfies the condition

where 1 is the constant function on V . This condition ensures that the gradient measures differences between values of the function across connected elements of the graph and vanishes for constant functions. If grad ( 1 ) ≠ 0 , the operator does not represent variations in the field and cannot be associated with a Laplacian of diffusive nature.

Proposition 7

The standard graph Laplacian L = K − A for an undirected graph is a Laplacian operator in the above sense.

Proof

Let ∇ denote the oriented vertex-edge incidence matrix of the graph, defined by

Define the discrete gradient as

so that ( grad x ) e = x head ( e ) − x tail ( e ) , and its adjoint (negative divergence) as

Then,

By construction,

so the gradient of a constant function vanishes. Therefore, L = ∇ ⊤ ∇ satisfies the defining property of a Laplacian operator.□

Proposition 8

On a directed graph G = ( V , E ) with adjacency matrix A, the out- and in-degree Laplacians

are not proper Laplacians, i.e., they cannot be written as L = div grad with a diffusive gradient grad satisfying grad ( 1 ) = 0 . Instead, they admit transport-type factorizations:

where U and V are vertex-edge and edge-vertex “tail” selectors (transport fields), and ∇ τ , div ρ are oriented discrete derivatives along edges. Hence, L out acts as an advection operator (outflow) and L in as its inflow/accumulation conjugate.

Proof

(1) Out- and in-degree Laplacians are not diffusive. Any operator of the form L = div grad with grad ( 1 ) = 0 and div = grad * is Hermitian and positive semidefinite. For a genuinely directed graph ( A ≠ A ⊤ ),

so it is neither self-adjoint nor diffusive in the standard inner product. Thus, they are not proper Laplacians.

(2) Advection factorization of L out . Choose an orientation of edges and let ∇ be the vertex-edge incidence (difference) matrix,

Let U be the vertex-edge “tail” selector,

Then, for any x ∈ C n ,

Hence,

which represents a discrete advection (outflow) operator with velocity field U .

(3) Inflow/accumulation factorization of L in . Define the discrete divergence div ≔ ∇ ⊤ and the edge-vertex “tail” selector

Then, componentwise

Reversing the orientation (equivalently, using the incidence of the reversed digraph) yields

so that

with div ρ the divergence in the reversed orientation. This operator aggregates incoming flux and thus acts as an inflow or accumulation operator – the natural conjugate of advection.□

Remark 9

The pair ( L out , L in ) realizes a discrete transport duality

Neither operator is a proper Laplacian, since they lack the diffusive-conservative form div grad with grad ( 1 ) = 0 ; they instead describe directed transport processes.

Proposition 10

Let G = ( V , E ) be a simple undirected graph with degree matrix K = diag ( k v ) , adjacency A, and identity I. For a real parameter r, define the deformed Laplacian

Then, except in two trivial regular cases, L ( r ) is neither diffusive (positive semidefinite) nor a proper Laplacian of the form L = div grad with a proper gradient satisfying grad ( 1 ) = 0 .

Proof

If L ( r ) was a proper Laplacian, the condition grad ( 1 ) = 0 would imply L ( r ) 1 = 0 . For each vertex v ,

For a fixed r , q v ( r ) = 0 for all v only if all degrees are equal ( G is k -regular), since q v depends on k v . In that case, the roots of ( k − 1 ) r 2 + k r + 1 = 0 are r = − 1 and r = − 1 ∕ ( k − 1 ) . Only for these two values do we have

which are (multiples of) the standard Laplacian and hence, proper in the strict diffusive-self-adjoint sense. For every other value of r , L ( r ) 1 ≠ 0 , so L ( r ) cannot be expressed as div grad with grad ( 1 ) = 0 .

Regarding diffusivity, for general r , the off-diagonal entries of L ( r ) on edges equal + r , which become positive when r > 0 . Consequently, x ⊤ L ( r ) x need not be nonnegative, and the matrix L ( r ) fails to be positive semidefinite except at the conservative points r = − 1 and r = − 1 ∕ ( k − 1 ) for regular graphs. In general, L ( r ) is indefinite and therefore non-diffusive. Hence, except for the two regular conservative cases, L ( r ) is neither diffusive nor a proper Laplacian.□

Remark 11

Whenever the diagonal quantities

are nonnegative (e.g., for r ≤ − 1 or for r ∈ [ − 1 ∕ ( Δ − 1 ) , 0 ] with Δ = max v k v ), one may write

Here ∇ is any oriented incidence matrix. This shows that L ( r ) is diffusive (positive semidefinite), but the added unary rows do not compare pairs of vertices, so the corresponding “gradient” does not annihilate constants. Therefore, even in this case, L ( r ) is not a proper Laplacian under the definition L = div grad with grad ( 1 ) = 0 .

Proposition 12

Let G = ( V , E ) be a directed or mixed graph with complex edge phases ω i j ∈ C satisfying ∣ ω i j ∣ = 1 and ω j i = ω i j ¯ whenever both orientations ( i , j ) and ( j , i ) are present. Define the magnetic (Hermitian) Laplacian

where K = diag ( k v ) is the degree matrix with k v = ∑ j ∣ ( A B ) v j ∣ . Then, L B is Hermitian and diffusive (it admits the factorization L B = ∇ B * ∇ B ), but it is not a proper Laplacian in the sense L = div grad with a proper gradient satisfying grad ( 1 ) = 0 , unless all phases are trivial ( ω i j ≡ 1 ). As a limiting real case, the signless Laplacian Q = K + A (corresponding to ω i j ≡ − 1 ) is also diffusive but not proper.

Proof

Fix an orientation of E and define the magnetic gradient ∇ B : C ( V ) → C ( E ) by

A direct computation yields the Hermitian diffusive factorization

However, for the constant vector 1 , we have

which is nonzero whenever ω i j ≠ 1 . Thus, ∇ B fails to be a proper gradient since it does not annihilate constants. Consequently, L B , while diffusive and self-adjoint, cannot be expressed as div grad with a proper gradient satisfying grad ( 1 ) = 0 unless all phases are trivial.

To exclude the possibility of another proper gradient ϒ generating L B , suppose L B = ϒ * ϒ = ∇ B * ∇ B . By the polar decomposition, there exists a partial isometry U such that ∇ B = U ϒ . Hence, ∇ B 1 = U ϒ 1 = 0 , contradicting the fact that ∇ B 1 ≠ 0 . Therefore, no proper gradient can yield L B unless ω i j ≡ 1 , in which case ∇ B reduces to the ordinary incidence matrix and L B becomes the standard Laplacian.

For the uniform phase ω i j ≡ − 1 , one has A B = − A and thus, L B = K − ( − A ) = K + A ≕ Q . The associated incidence operator is

for which ∇ + 1 ≠ 0 . Repeating the same argument shows that no proper gradient ϒ with ϒ 1 = 0 can satisfy Q = ϒ * ϒ . Hence, Q is also diffusive but not proper.□

Remark 13

The magnetic Laplacian L B can be interpreted as the Schur complement of a proper Laplacian on an extended vertex-edge space. Given an oriented or mixed graph G = ( V , E ) and magnetic phases ω i j with ∣ ω i j ∣ = 1 , introduce auxiliary edge variables z e ∈ C and define the extended gradient

In block form,

The corresponding extended Laplacian

is Hermitian positive semidefinite and of the diffusive form grad * grad on the extended space V ∪ E . Eliminating the edge variables z gives the Schur complement

Thus, the magnetic Laplacian arises as the projection of a proper Laplacian defined on a higher-dimensional space. The signless Laplacian Q = K + A corresponds to the real case with a uniform magnetic phase ω i j = − 1 (a phase shift of π ) along all oriented edges.

Example 14

Consider the undirected 3-cycle with edges { 1 , 2 } , { 2 , 3 } , { 3 , 1 } and magnetic phases ω 12 , ω 23 , ω 31 ∈ C satisfying ∣ ω i j ∣ = 1 and ω j i = ω i j ¯ . The magnetic Laplacian is

We introduce one auxiliary variable z i j for each oriented edge ( i → j ) ∈ { ( 1 , 2 ) , ( 2 , 3 ) , ( 3 , 1 ) } . Define the extended magnetic gradient ϒ : C 3 × C 3 → C 6 by

In matrix form, this can be written as

where

Hence,

which for the example in question is given by

Eliminating the edge variables z gives the Schur complement

Therefore, the Hermitian magnetic Laplacian L B is precisely the Schur complement of the proper extended Laplacian ℒ ext = ϒ * ϒ . Note that in this particular example, due to the specific choice of orientation and normalization, L B is also the upper-left block of ℒ ext .

Remark 15

Let L B = K − A B be the magnetic Laplacian as defined before, such that A B ( i , j ) = ω i j ∈ C if ( i , j ) ∈ E and ω j i = ω i j ¯ . Let ϕ = Re ( ω i j ) and let φ = Im ( ω i j ) . Let R be the matrix whose entries are defined as

Then, we can write

Let us plug this operator on the diffusion equation:

Then, L ϕ is a diffusive Laplacian with diffusion coefficient ϕ and i R can be considered as an imaginary reaction potential, which account for wave-like reactions. The equation is similar to the one proposed by Varshney et al. [85] to model a diffusive process plus a chemical potential in neuronal systems. The difference is that the model of Varshney et al. [85] does not include the term φ i multiplying the chemical potential, which makes that their operator is not symmetric, its eigenvalues may be complex, and it could be nondiagonalizable. Additionally, the semigroup obtained from their model is neither stochastic nor sub-Markovian.

Proposition 16

Let G σ = ( V , E , σ ) be a signed graph with edge signs σ i j ∈ { ± 1 } on { i , j } ∈ E , and let

Then, L σ is Hermitian and diffusive (it admits the factorization L σ = ∇ σ * ∇ σ ), but it is not a proper Laplacian in the sense L = div grad with a proper gradient satisfying grad ( 1 ) = 0 , unless the signing is trivial ( σ i j ≡ + 1 on all edges).

Proof

Fix an orientation of E and define the signed (“twisted”) gradient ∇ σ : C ( V ) → C ( E ) by

A direct computation gives the standard factorization

Evaluating this operator on the constant vector 1 yields

Hence, ∇ σ 1 = 0 if and only if σ i j = + 1 on every edge. Whenever a negative edge is present, ∇ σ fails to be a proper gradient, since it does not annihilate constants.

Suppose there existed another proper gradient ϒ with ϒ 1 = 0 such that L σ = ϒ * ϒ = ∇ σ * ∇ σ . Then, by the polar decomposition, there would exist a partial isometry U such that ∇ σ = U ϒ . Applying both sides 1 gives ∇ σ 1 = U ϒ 1 = 0 , a contradiction when any negative edge exists. Thus, L σ can only be proper when all σ i j = + 1 , in which case it reduces to the standard Laplacian.□

Remark 17

The signed Laplacian L σ = D − A σ can be decomposed as the difference of two proper Laplacians,

where L + and L − are the ordinary Laplacians of the subgraphs consisting of positive and negative edges, respectively. Hence, L σ represents the superposition of two diffusive processes of opposite sign: one promoting alignment across positive links, and the other promoting anti-alignment across negative ones. It is therefore not a purely diffusive operator but the generator of a competition between cooperative and antagonistic diffusion.

Proposition 18

Let G = ( V , E ) be a (simple) undirected graph with degree matrix K = diag ( k 1 , … , k n ) and standard incidence (edge-vertex) matrix ∇ (so L = ∇ ∇ ⊤ = K − A ). Fix a parameter χ ∈ R and define the augmented (“hairy”) incidence

interpreting each diagonal entry of ℛ as a semiedge connecting vertex i to an external reservoir. Then, the Lerman-Ghosh operator

is self-adjoint and positive semidefinite (diffusive) and satisfies

However, L χ is not a proper Laplacian in the sense L = grad * grad with a proper gradient such that grad ( 1 ) = 0 .

Proof

Block multiplication gives

where ℛ 2 = diag ( max { χ − k i , 0 } ) . Since L χ is a Gram operator, it is symmetric and positive semidefinite: