Laplacian state transfer in graphs with involutions

Hermie Monterde; Hiranmoy Pal; Swornalata Ojha

arxiv: 2604.20700 · v1 · submitted 2026-04-22 · 🧮 math.CO

Laplacian state transfer in graphs with involutions

Swornalata Ojha , Hermie Monterde , Hiranmoy Pal This is my paper

Pith reviewed 2026-05-10 00:02 UTC · model grok-4.3

classification 🧮 math.CO

keywords perfect state transfergeneralized Laplaciangraphs with involutionsplanar graphstreespathscyclespair states

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The pith

Perfect state transfer between pair states in a graph with involution is equivalent to transfer between vertices in the induced subgraph under the generalized Laplacian.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that in graphs with a non-trivial involution, the existence of perfect state transfer between certain pair or plus states using the generalized Laplacian as Hamiltonian is equivalent to perfect state transfer between vertices in the subgraph induced by the involution. This equivalence is applied to show that adding loops of weight one to exactly two vertices produces such transfer in almost all simple unweighted planar graphs and almost all simple unweighted trees. The work also determines exact conditions for end-to-end transfer on paths, shows that adding one edge to a cycle does not produce transfer under Laplacian or signless Laplacian Hamiltonians, and constructs examples of PST in modified complete bipartite graphs, cycles, and paths.

Core claim

We establish an equivalence between the existence of PST between certain pair (or plus states) in such a graph and PST between vertices in a subgraph induced by the involution. This allows us to prove that for almost all simple unweighted planar graphs (resp., almost all simple unweighted trees), the assignment of loops of weight one to exactly two vertices in the graph produces PST between pair states relative to ℒ. We also show that a path on n vertices admits PST between end vertices relative to ℒ if and only if n=2, or (n,q)=(3,(k²-l²)/8l²) where k>l are integers with k ≢ l mod 2. For cycles, the addition of an extra edge does not yield PST between vertices relative to Laplacian and sign

What carries the argument

The equivalence between PST on pair or plus states in the full graph and vertex PST in the involution-induced subgraph, obtained from the action of the non-trivial involution on the spectrum of the generalized Laplacian ℒ = Δ + qA.

If this is right

Almost all simple unweighted planar graphs admit PST between pair states after adding two weight-one loops.
Almost all simple unweighted trees admit PST between pair states after adding two weight-one loops.
A path on n vertices has PST between its end vertices if and only if n equals 2 or n equals 3 with q equal to (k squared minus l squared) over 8 l squared for integers k greater than l of opposite parity.
Adding a single extra edge to any cycle fails to produce PST between vertices under either the Laplacian or signless Laplacian.
Suitable additions of edges and loops create PST between pair states in complete bipartite graphs, cycles, and paths.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reduction via involution could be used to search for new families of graphs admitting PST by first solving the problem on smaller symmetric subgraphs.
The same symmetry argument might apply when the Hamiltonian is replaced by other matrices that commute with the involution action.
Explicit constructions for paths and trees suggest that most graphs in those classes can be made to support PST with minimal local modifications.

Load-bearing premise

The graphs are simple and unweighted with a non-trivial involution, and the generalized Laplacian serves as the Hamiltonian whose eigenvalues govern the state transfer.

What would settle it

A concrete graph equipped with a non-trivial involution in which pair-state PST occurs under ℒ but vertex PST fails in the induced subgraph (or the converse) would disprove the equivalence.

Figures

Figures reproduced from arXiv: 2604.20700 by Hermie Monterde, Hiranmoy Pal, Swornalata Ojha.

**Figure 2.** Figure 2: Cycles with tails with pair state transfer between [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: Graphs X (left) and Y (right) with involution ϕ = (a c)(b d) that fixes H In [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: Planar graphs with plus PST between √ 1 2 (ea + ec) and √ 1 2 (eb + ed) relative to L at π 2q in (i), π 2q in (ii) and √π 2q in (iii); the edges of the half graphs are drawn in red. Remark 3. If G in Theorem 5 is a regular graph, then for the theorem to hold, we only need to assign loops of weight 1 + q to vertices in X that are not matched in Mk. Example 2. The three graphs in [PITH_FULL_IMAGE:figures/fu… view at source ↗

**Figure 5.** Figure 5: K3,3−Mk and K4,3−Mk+E with pair PST between √ 1 2 (e1−e3) and √ 1 2 (e2−e4). connecting each of the vertices a and b to m−n additional vertices, chosen to preserve the involution ϕ. The same argument yields pair PST relative to the generalized Laplacian matrix of Km,n − Mk + E at π 2q between the same pair states (see right of [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: Pair PST on cycles with loops and additional edges. [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗

read the original abstract

For $q\in\mathbb{R}\backslash\{0\}$, the generalized Laplacian of a graph $X$ is the matrix $\mathscr{L}=\Delta+qA$, where $\Delta$ is the degree matrix and $A$ is the adjacency matrix of $X$. In this paper, we investigate perfect state transfer (PST) on graphs with possible loops equipped with non-trivial involutions, where we take the generalized Laplacian matrix as the Hamiltonian of the underlying spin network. We establish an equivalence between the existence of PST between certain pair (or plus states) in such a graph and PST between vertices in a subgraph induced by the involution. This allows us to prove that for almost all simple unweighted planar graphs (resp., almost all simple unweighted trees), the assignment of loops of weight one to exactly two vertices in the graph produces PST between pair states relative to $\mathscr{L}$. We also show that a path on $n$ vertices admits PST between end vertices relative to $\mathscr{L}$ if and only if $n =2$, or $(n,q)=(3,\frac{k^2-l^2}{8l^2})$ where $k>l$ are integers with $k \not\equiv l \pmod{2}$. For cycles, we show that the addition of an extra edge does not yield PST between vertices relative to Laplacian and signless Laplacian matrices. Furthermore, we show that the addition of a few suitable edges (including loops) in complete bipartite graphs, cycles, and paths yields PST between pair states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The involution-reduction equivalence is the clean new piece here, but the 'almost all' claim for planar graphs and trees looks underspecified on how the involution arises.

read the letter

The paper reduces PST between pair or plus states in a graph with a non-trivial involution to ordinary PST between vertices in the induced subgraph on the orbits. That equivalence is the main new tool, and it lets them handle the generalized Laplacian with the parameter q. They also give an exact condition for when a path has PST between ends: only for n=2 or the specific rational q when n=3. The constructions that add a few edges or loops to cycles, paths, and complete bipartite graphs to create PST are straightforward extensions of earlier work on standard Laplacians. The cycle result that an extra edge fails to produce PST is a small negative observation that rules out one simple modification. Overall the derivations stay within algebraic graph theory and spectral arguments on symmetric matrices, with no obvious circularity or fitted parameters beyond the free q. The path characterization is concrete enough that a reader can check it directly. The soft spot is the density claim. The abstract says that assigning loops of weight one to exactly two vertices produces PST between pair states for almost all simple unweighted planar graphs and almost all trees. But pair states and the reduction both require a non-trivial involution to be present. Most planar graphs and trees have trivial automorphism groups, so the set of graphs that already admit such an involution has density zero. If the paper instead means that for almost all graphs one can always find two vertices whose loops create a suitable involution, then the argument needs an explicit construction that works generically rather than a post-hoc choice. The abstract does not resolve which reading is intended, and that leaves the main existence statement hanging. This work is aimed at people already working on perfect state transfer in graphs or on quantum walks with algebraic Hamiltonians. A specialist can pull the equivalence and the path condition for their own constructions. The rest is incremental. I would send it to a referee because the reduction is worth verifying and the path theorem is self-contained, even though the quantification on planar graphs and trees needs a clearer statement or a fix.

Referee Report

2 major / 3 minor

Summary. The paper investigates perfect state transfer (PST) on graphs with non-trivial involutions, taking the generalized Laplacian ℒ = Δ + qA (q ≠ 0) as the Hamiltonian. It establishes an equivalence between PST on pair (or plus) states in such a graph and PST between vertices in the subgraph induced by the involution. This reduction is applied to prove that assigning loops of weight one to exactly two vertices produces PST between pair states for almost all simple unweighted planar graphs and almost all simple unweighted trees. Additional results include an if-and-only-if characterization of PST between endpoints of a path on n vertices (n=2 or the specific rational condition on q for n=3), non-existence of PST after adding an edge to cycles for both Laplacian and signless Laplacian, and constructions yielding PST between pair states in complete bipartite graphs, cycles, and paths via added edges or loops.

Significance. If the equivalence holds, the reduction to the quotient graph induced by the involution is a useful structural tool for studying PST in symmetric graphs and could streamline proofs for large families. The application to 'almost all' planar graphs and trees, if the quantification is made precise, would show that simple local modifications suffice to produce PST in a dense class of graphs, which is of interest in algebraic graph theory and quantum information. The explicit path characterization and the negative results for cycles add concrete, checkable statements to the literature on Laplacian-based state transfer.

major comments (2)

[Abstract] Abstract and the main application section: the claim that adding loops of weight 1 to exactly two vertices yields PST between pair states 'for almost all simple unweighted planar graphs (resp., almost all simple unweighted trees)' is load-bearing but ambiguous. Pair states and the equivalence are defined only once a non-trivial involution σ is fixed, yet the set of planar graphs or trees admitting a non-trivial automorphism has density zero. The manuscript must clarify whether 'almost all' is taken inside the subclass of graphs that already possess such an involution, or whether the two vertices are chosen so as to induce a suitable σ (with an explicit construction).
[Equivalence theorem] Equivalence theorem (the section establishing the reduction to the induced subgraph on orbits/fixed points): the argument must explicitly verify that the generalized Laplacian with the added loops remains compatible with the involution action, i.e., that the loop weights are placed symmetrically or that the quotient construction accounts for the diagonal perturbation without introducing extra eigenvalues that break the PST equivalence.

minor comments (3)

[Notation] Notation section: define 'pair states' and 'plus states' explicitly at the first appearance, including their relation to the eigenspaces of the involution operator, rather than assuming familiarity from prior PST literature.
[Path theorem] Path theorem: the condition '(n,q)=(3,(k²-l²)/(8l²)) with k>l integers, k ≢ l mod 2' should be accompanied by a brief remark on whether this exhausts all real q or only produces PST for those discrete values; also confirm that the spectral argument rules out other n.
[Constructions for cycles and K_{m,n}] Cycle and complete-bipartite constructions: the added edges/loops are described as 'a few suitable' ones; a table or explicit listing of the minimal number and placement for each family would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract and the main application section: the claim that adding loops of weight 1 to exactly two vertices yields PST between pair states 'for almost all simple unweighted planar graphs (resp., almost all simple unweighted trees)' is load-bearing but ambiguous. Pair states and the equivalence are defined only once a non-trivial involution σ is fixed, yet the set of planar graphs or trees admitting a non-trivial automorphism has density zero. The manuscript must clarify whether 'almost all' is taken inside the subclass of graphs that already possess such an involution, or whether the two vertices are chosen so as to induce a suitable σ (with an explicit construction).

Authors: We agree the quantification is ambiguous as written and will revise the abstract and application section for precision. The intended meaning is that the result holds for almost all graphs (in the asymptotic-density sense within the class) that already admit a fixed non-trivial involution σ; the two vertices receiving loops are then chosen symmetrically with respect to that σ (e.g., a swapped pair or a pair of fixed points) so that the modified graph remains σ-invariant. We will add an explicit construction paragraph showing how, given σ, the vertices are selected and why this produces the required pair states. We also note that the absolute density among all planar graphs or trees is indeed zero, and the revised statement will make the relative quantification explicit. revision: yes
Referee: [Equivalence theorem] Equivalence theorem (the section establishing the reduction to the induced subgraph on orbits/fixed points): the argument must explicitly verify that the generalized Laplacian with the added loops remains compatible with the involution action, i.e., that the loop weights are placed symmetrically or that the quotient construction accounts for the diagonal perturbation without introducing extra eigenvalues that break the PST equivalence.

Authors: The loops are placed symmetrically by construction: either on a σ-swapped pair (so each receives the same weight) or on fixed points whose loop contributions are absorbed into the orbit degrees. Consequently the modified ℒ commutes with the linear action of σ, the symmetric/antisymmetric decomposition of the space is preserved, and the quotient Laplacian on the orbit graph exactly encodes the diagonal perturbation without extraneous eigenvalues. We will insert a short verification paragraph (or lemma) right after the equivalence theorem statement that spells out this invariance and confirms the PST reduction remains exact. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives an equivalence between PST on pair/plus states in a graph with involution and PST on the induced subgraph on orbits/fixed points, using the spectral decomposition of the generalized Laplacian ℒ under the involution action. This equivalence is obtained directly from the commutation relations and eigenspace projections of ℒ with the involution operator, without reducing to fitted parameters or self-referential definitions. The subsequent claim about 'almost all' planar graphs and trees follows from applying the equivalence after choosing two vertices to induce loops and an involution, relying on standard density arguments in the space of graphs rather than any input data or prior self-citation that is load-bearing. No ansatz is smuggled, no uniqueness theorem is imported from the authors' prior work, and no known result is merely renamed. The derivation chain remains independent of its target conclusions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The paper rests on standard definitions and spectral theory of graphs and matrices; no new entities are postulated and q is treated as a free real parameter.

free parameters (1)

q
Real nonzero scalar scaling the adjacency matrix in the generalized Laplacian; specific values are solved for in the path case.

axioms (2)

standard math Standard spectral properties of the Laplacian and adjacency matrices under group actions (involutions).
Invoked throughout to relate eigenvalues of the full graph to the quotient or induced subgraph.
domain assumption Graphs are finite, simple, and unweighted except for the added loops of weight one.
Used to state the 'almost all' results for planar graphs and trees.

pith-pipeline@v0.9.0 · 5576 in / 1419 out tokens · 38208 ms · 2026-05-10T00:02:32.818653+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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Chen and C

Q. Chen and C. Godsil. Pair state transfer.Quantum Inf. Process., 19(9):321, 2020

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Christandl, N

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C. Godsil. State transfer on graphs.Discrete Math., 312(1):129–147, 2012

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Godsil, S

C. Godsil, S. Kirkland, S. Mohapatra, H. Monterde, and H. Pal. Quantum walks on finite and bounded infinite graphs.arXiv preprint arXiv:2510.05306, 2025

work page arXiv 2025
[7]

Godsil, S

C. Godsil, S. Kirkland, and H. Monterde. Perfect state transfer between real pure states.SIAM J. Matrix Anal. Appl., 46(3):2093–2115, 2025. 22

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A. Kay. Perfect, efficient, state transfer and its application as a constructive tool. Int. J. Quantum Inf., 8(04):641–676, 2010

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Kempton, G

M. Kempton, G. Lippner, and S. Yau. Perfect state transfer on graphs with a potential.Quantum Inf. Comput., 17(3):303–327, 2017

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Kempton, G

M. Kempton, G. Lippner, and S.-T. Yau. Pretty good quantum state transfer in symmetric spin networks via magnetic field.Quantum Inf. Process., 16:1–23, 2017

work page 2017
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S. Kim, H. Monterde, B. Ahmadi, A. Chan, S. Kirkland, and S. Plosker. A general- ization of quantum pair state transfer.Quantum Inf. Process., 23(11):369, 2024

work page 2024
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Kirkland, H

S. Kirkland, H. Monterde, and S. Plosker. Quantum state transfer between twins in weighted graphs.J. Algebraic Combin., 58(3):623–649, 2023

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Monterde

H. Monterde. Sedentariness in quantum walks.Quantum Inf. Process., 22(7):273, 2023

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[14]

Monterde and H

H. Monterde and H. Pal. Perfect state transfer on graphs with clusters.arXiv preprint arXiv:2505.07982, 2025

work page arXiv 2025
[15]

Pal and B

H. Pal and B. Bhattacharjya. Pretty good state transfer on some neps.Discrete Math., 340(4):746–752, 2017

work page 2017
[16]

Pal and S

H. Pal and S. Mohapatra. Quantum pair state transfer on isomorphic branches. Linear Algebra Appl., 740:151–172, 2026

work page 2026
[17]

W. So. Rank one perturbation and its application to the laplacian spectrum of a graph.Linear Multilinear Algebra, 46(3):193–198, 1999

work page 1999
[18]

Vinet and A

L. Vinet and A. Zhedanov. Almost perfect state transfer in quantum spin chains. Phys. Rev. A, 86(5):052319, 2012. 23

work page 2012

[1] [1]

Chen and C

Q. Chen and C. Godsil. Pair state transfer.Quantum Inf. Process., 19(9):321, 2020

work page 2020

[2] [2]

Christandl, N

M. Christandl, N. Datta, A. Ekert, and A. J. Landahl. Perfect state transfer in quantum spin networks.Phys. Rev. Lett., 92:187902, May 2004

work page 2004

[3] [3]

Coutinho, E

G. Coutinho, E. Juliano, and T. J. Spier. No perfect state transfer in trees with more than 3 vertices.J. Combin. Theory Ser. B, 168:68–85, 2024

work page 2024

[4] [4]

C. Godsil. State transfer on graphs.Discrete Math., 312(1):129–147, 2012

work page 2012

[5] [5]

C. Godsil. When can perfect state transfer occur?Electron. J. Linear Algebra, 23:877–890, 2012

work page 2012

[6] [6]

Godsil, S

C. Godsil, S. Kirkland, S. Mohapatra, H. Monterde, and H. Pal. Quantum walks on finite and bounded infinite graphs.arXiv preprint arXiv:2510.05306, 2025

work page arXiv 2025

[7] [7]

Godsil, S

C. Godsil, S. Kirkland, and H. Monterde. Perfect state transfer between real pure states.SIAM J. Matrix Anal. Appl., 46(3):2093–2115, 2025. 22

work page 2093

[8] [8]

A. Kay. Perfect, efficient, state transfer and its application as a constructive tool. Int. J. Quantum Inf., 8(04):641–676, 2010

work page 2010

[9] [9]

Kempton, G

M. Kempton, G. Lippner, and S. Yau. Perfect state transfer on graphs with a potential.Quantum Inf. Comput., 17(3):303–327, 2017

work page 2017

[10] [10]

Kempton, G

M. Kempton, G. Lippner, and S.-T. Yau. Pretty good quantum state transfer in symmetric spin networks via magnetic field.Quantum Inf. Process., 16:1–23, 2017

work page 2017

[11] [11]

S. Kim, H. Monterde, B. Ahmadi, A. Chan, S. Kirkland, and S. Plosker. A general- ization of quantum pair state transfer.Quantum Inf. Process., 23(11):369, 2024

work page 2024

[12] [12]

Kirkland, H

S. Kirkland, H. Monterde, and S. Plosker. Quantum state transfer between twins in weighted graphs.J. Algebraic Combin., 58(3):623–649, 2023

work page 2023

[13] [13]

Monterde

H. Monterde. Sedentariness in quantum walks.Quantum Inf. Process., 22(7):273, 2023

work page 2023

[14] [14]

Monterde and H

H. Monterde and H. Pal. Perfect state transfer on graphs with clusters.arXiv preprint arXiv:2505.07982, 2025

work page arXiv 2025

[15] [15]

Pal and B

H. Pal and B. Bhattacharjya. Pretty good state transfer on some neps.Discrete Math., 340(4):746–752, 2017

work page 2017

[16] [16]

Pal and S

H. Pal and S. Mohapatra. Quantum pair state transfer on isomorphic branches. Linear Algebra Appl., 740:151–172, 2026

work page 2026

[17] [17]

W. So. Rank one perturbation and its application to the laplacian spectrum of a graph.Linear Multilinear Algebra, 46(3):193–198, 1999

work page 1999

[18] [18]

Vinet and A

L. Vinet and A. Zhedanov. Almost perfect state transfer in quantum spin chains. Phys. Rev. A, 86(5):052319, 2012. 23

work page 2012