Unlabeled Quantum Graphs

• Unlabeled quantum graphs are quantum representations in which vertex identities are not fixed, allowing analysis of permutation invariant states and novel symmetry properties.
• They employ approaches like permutation-invariant Hilbert spaces, quantum adjacency subspaces, and configuration spaces to study graph isomorphisms and thermodynamic phase transitions.
• These formalisms connect quantum statistics, spectral graph theory, and noncommutative geometry, offering actionable insights for advancing quantum network and symmetry research.

Unlabeled quantum graphs are graph-theoretic quantum objects in which vertex identities are not treated as primitive observables. In the current literature, this designation covers several distinct constructions rather than a single formalism: permutation-invariant Hilbert spaces of finite multigraphs with indistinguishable vertices, quantum adjacency subspaces $\mathcal U_G$ attached to classical graphs, configuration spaces $C_n(X)=(X^{\times n}-\Delta_n)/S_n$ for indistinguishable particles on a graph, and quantum graphs considered only up to quantum isomorphism (Betre et al., 10 Sep 2025, Ostrovska et al., 10 Mar 2026, Maciążek et al., 2018, Matsuda, 2021). Across these settings, unlabeledness changes the status of symmetry, locality, and reconstruction: it can turn isomorphism classes into physical basis states, enlarge automorphism structures beyond classical graph symmetries, and require a careful distinction between quotienting by labels and imposing renaming invariance as a symmetry principle (Betre et al., 10 Sep 2025, Ostrovska et al., 10 Mar 2026, Arrighi et al., 2020).

1. Multiple formalizations of unlabeledness

A first source of ambiguity is that “unlabeled” refers to different operations in different subfields. In the finite-multigraph formalism, unlabeledness is implemented by restricting to the $S_N$-invariant subspace of a labeled graph Hilbert space, so that physical states are permutation-invariant superpositions corresponding to graph isomorphism classes (Betre et al., 10 Sep 2025). In the operator-algebraic theory of quantum symmetries, a simple finite graph $G$ is replaced by the quantum adjacency subspace

$$\mathcal{U}_G=\operatorname{span}\{e_x\otimes e_y:x\sim_G y\}\subseteq \mathbb{C}^{V(G)}\otimes \mathbb{C}^{V(G)},$$

and unlabeledness is studied through the game algebra $C(\mathrm{Qut}(\mathcal U_G))$ rather than through permutations of a vertex list (Ostrovska et al., 10 Mar 2026). In the topology of indistinguishable particles, unlabeledness means quotienting the configuration space by the symmetric group,

$C_n(X):=(X^{\times n}-\Delta_n)/S_n,$

so that particle permutations are removed from the classical configuration data (Maciążek et al., 2018). In quantum graph isomorphism theory, unlabeled quantum graphs are equivalence classes modulo quantum isomorphisms, not merely classical relabelings (Matsuda, 2021).

Formal setting	Basic object	Mode of unlabeledness
Finite quantum multigraphs	$S_N$-invariant subspace of $\mathcal H_{MG}$	Physical states are permutation-invariant orbit sums
Quantum adjacency subspaces	$$\mathcal U_G\subseteq \mathbb C^{V(G)}\otimes \mathbb C^{V(G)}$$	Symmetry encoded by a quantum automorphism game algebra
Indistinguishable particles on graphs	$C_n(X)=(X^{\times n}-\Delta_n)/S_n$0	Particle permutations are quotiented out
Quantum graphs up to quantum isomorphism	$C_n(X)=(X^{\times n}-\Delta_n)/S_n$1 or $C_n(X)=(X^{\times n}-\Delta_n)/S_n$2	Equivalence is by quantum bijections/intertwiners

These formalisms are related but not interchangeable. A plausible implication is that “unlabeled quantum graph” is best read as a family resemblance term: the common feature is removal or weakening of fixed vertex identity, while the mathematical implementation depends on whether the primary object is a Hilbert space, a $C_n(X)=(X^{\times n}-\Delta_n)/S_n$3-algebra, a configuration space, or a symmetry game.

2. Permutation-invariant quantum multigraphs

For finite undirected multigraphs with no self-loops, the labeled Hilbert space is

$C_n(X)=(X^{\times n}-\Delta_n)/S_n$4

Unlabeled quantum multigraphs are obtained by passing to the $C_n(X)=(X^{\times n}-\Delta_n)/S_n$5-invariant subspace,

$C_n(X)=(X^{\times n}-\Delta_n)/S_n$6

and a physical state is the symmetrization of a labeled state,

$C_n(X)=(X^{\times n}-\Delta_n)/S_n$7

In the labeled case all $C_n(X)=(X^{\times n}-\Delta_n)/S_n$8 states are distinct; in the unlabeled case the physical states are $C_n(X)=(X^{\times n}-\Delta_n)/S_n$9-orbits, interpreted as isomorphism classes of graphs (Betre et al., 10 Sep 2025).

This change of kinematics has direct thermodynamic consequences. The free theory of labeled quantum simple graphs is the Erdős–Rényi–Gilbert $S_N$0 model of random graphs, with analytic free energy and no thermodynamic phase transition. By contrast, unlabeled quantum graphs exhibit proper thermodynamic phase transitions in both the free and ferromagnetic Ising models, with divergence in the specific heat, divergence in the susceptibility, and critical slowing near the critical temperature (Betre et al., 10 Sep 2025). The order parameter is the fraction of vertices in the largest connected component,

$S_N$1

with susceptibility

$S_N$2

As temperature decreases, the graphs transition from having a non-trivial automorphism group to a trivial one, corresponding to the appearance of a giant connected component (Betre et al., 10 Sep 2025).

The formal interpretation given for this construction is explicitly relational. Unlabeled graphs model physically identical constituents, and the quantization is described as a quantum mechanical treatment of the relations themselves rather than of particles living on a fixed graph. In the unlabeled formalism, local vertex observables are projected out and only global, symmetric operators survive (Betre et al., 10 Sep 2025). This suggests a strong distinction between vertex-indistinguishability as a dynamical principle and vertex anonymity as a purely combinatorial quotient.

3. Quantum adjacency subspaces and nonlocal symmetry

Given a simple finite undirected graph $S_N$3, the associated quantum graph

$S_N$4

embeds $S_N$5 into a broader theory of quantum graphs where adjacency is encoded by a symmetric subspace rather than by a $S_N$6-$S_N$7 matrix alone (Ostrovska et al., 10 Mar 2026). Its quantum automorphism structure is encoded by the game algebra $S_N$8, whose tracial states correspond to perfect quantum no-signaling strategies for the automorphism game. The algebra is generated by expressions of the form $S_N$9, where $G$0 is bi-unitary and the relations enforce adjacency preservation (Ostrovska et al., 10 Mar 2026).

For complete graphs $G$1, the unlabeled quantum symmetry is strictly richer than the labeled one. There is an injective homomorphism $G$2, but noncommutativity appears already for $G$3, whereas in the labeled case $G$4 is first noncommutative at $G$5 (Ostrovska et al., 10 Mar 2026). There is also a surjective $G$6-homomorphism from $G$7 onto $G$8, which exhibits substantial noncommutative structure (Ostrovska et al., 10 Mar 2026).

The most global statement is that every graph with at least three vertices gives rise to an unlabeled quantum graph admitting nonlocal symmetry: for any $G$9 with $$\mathcal{U}_G=\operatorname{span}\{e_x\otimes e_y:x\sim_G y\}\subseteq \mathbb{C}^{V(G)}\otimes \mathbb{C}^{V(G)},$$0, $$\mathcal{U}_G=\operatorname{span}\{e_x\otimes e_y:x\sim_G y\}\subseteq \mathbb{C}^{V(G)}\otimes \mathbb{C}^{V(G)},$$1 has a perfect quantum no-signaling symmetry that is not local (Ostrovska et al., 10 Mar 2026). In the complete-graph case this is sharply different from the classical situation, where $$\mathcal{U}_G=\operatorname{span}\{e_x\otimes e_y:x\sim_G y\}\subseteq \mathbb{C}^{V(G)}\otimes \mathbb{C}^{V(G)},$$2 has nonlocal symmetry only for $$\mathcal{U}_G=\operatorname{span}\{e_x\otimes e_y:x\sim_G y\}\subseteq \mathbb{C}^{V(G)}\otimes \mathbb{C}^{V(G)},$$3. A plausible implication is that unlabeled quantum symmetries are not merely refinements of classical automorphisms but genuinely new symmetry resources tied to noncommutativity and nonlocal correlations.

4. Quantum isomorphism classes and explicit classifications

In operator-algebraic quantum graph theory, a quantum graph may be described as a pair $$\mathcal{U}_G=\operatorname{span}\{e_x\otimes e_y:x\sim_G y\}\subseteq \mathbb{C}^{V(G)}\otimes \mathbb{C}^{V(G)},$$4, with $$\mathcal{U}_G=\operatorname{span}\{e_x\otimes e_y:x\sim_G y\}\subseteq \mathbb{C}^{V(G)}\otimes \mathbb{C}^{V(G)},$$5 a finite quantum set and $$\mathcal{U}_G=\operatorname{span}\{e_x\otimes e_y:x\sim_G y\}\subseteq \mathbb{C}^{V(G)}\otimes \mathbb{C}^{V(G)},$$6 an orthogonal projection, or equivalently by an adjacency operator $$\mathcal{U}_G=\operatorname{span}\{e_x\otimes e_y:x\sim_G y\}\subseteq \mathbb{C}^{V(G)}\otimes \mathbb{C}^{V(G)},$$7 obtained by rotation (Gromada, 2021). A related formulation uses a quantum set $$\mathcal{U}_G=\operatorname{span}\{e_x\otimes e_y:x\sim_G y\}\subseteq \mathbb{C}^{V(G)}\otimes \mathbb{C}^{V(G)},$$8 and a quantum adjacency operator $$\mathcal{U}_G=\operatorname{span}\{e_x\otimes e_y:x\sim_G y\}\subseteq \mathbb{C}^{V(G)}\otimes \mathbb{C}^{V(G)},$$9 satisfying Schur idempotence,

$C(\mathrm{Qut}(\mathcal U_G))$0

with additional conditions for self-adjointness, reality, and reflexivity (Matsuda, 2021). In this framework, unlabeled quantum graphs are classes modulo quantum isomorphism. A quantum isomorphism between $C(\mathrm{Qut}(\mathcal U_G))$1 and $C(\mathrm{Qut}(\mathcal U_G))$2 is a quantum bijection $C(\mathrm{Qut}(\mathcal U_G))$3 satisfying

$C(\mathrm{Qut}(\mathcal U_G))$4

(Matsuda, 2021).

Two classification results on $C(\mathrm{Qut}(\mathcal U_G))$5 are especially concrete. For simple quantum graphs over $C(\mathrm{Qut}(\mathcal U_G))$6, the isomorphism class is determined solely by the number of quantum edges $C(\mathrm{Qut}(\mathcal U_G))$7, so up to isomorphism there are exactly four cases (Gromada, 2021). For undirected reflexive quantum graphs on $C(\mathrm{Qut}(\mathcal U_G))$8, every such graph is, up to quantum and classical isomorphism, one of four graphs parameterized by regular degree $C(\mathrm{Qut}(\mathcal U_G))$9; these are mutually non-isomorphic and all are classical (Matsuda, 2021). The latter result comes with explicit quantum isomorphisms to classical graphs on four vertices and reproduces monoidal equivalences between $C_n(X):=(X^{\times n}-\Delta_n)/S_n,$0 and $C_n(X):=(X^{\times n}-\Delta_n)/S_n,$1, and between $C_n(X):=(X^{\times n}-\Delta_n)/S_n,$2 and $C_n(X):=(X^{\times n}-\Delta_n)/S_n,$3 (Matsuda, 2021).

The same literature also shows that unlabeled quantum graphs need not have classical representatives. Twisting Cayley graphs of finite abelian groups by 2-cocycles produces quantum graphs that remain quantum isomorphic to the original classical graphs, such as the anticommutative hypercube built from the Clifford algebra $C_n(X):=(X^{\times n}-\Delta_n)/S_n,$4 (Gromada, 2021). By contrast, a quantum graph over $C_n(X):=(X^{\times n}-\Delta_n)/S_n,$5 with adjacency $C_n(X):=(X^{\times n}-\Delta_n)/S_n,$6 has a noncommutative Schur endomorphism algebra and is not quantum isomorphic to any classical graph (Gromada, 2021). Unlabeledness, in this sense, is therefore stronger than forgetting names: it organizes a landscape in which some equivalence classes contain classical graphs and others are intrinsically nonclassical.

5. Indistinguishable particles, configuration spaces, and statistics

A different but influential notion of unlabeledness arises when the graph is the one-particle configuration space and indistinguishable particles are quantized on it. The $C_n(X):=(X^{\times n}-\Delta_n)/S_n,$7-particle configuration space is

$C_n(X):=(X^{\times n}-\Delta_n)/S_n,$8

and quantum statistics are classified by unitary representations of its braid group,

$C_n(X):=(X^{\times n}-\Delta_n)/S_n,$9

up to conjugation (Maciążek et al., 2018). The moduli space of flat $S_N$0-bundles is

$S_N$1

so homology and cohomology of $S_N$2 control the possible quantum statistics (Maciążek et al., 2018).

For two indistinguishable spinless particles on a combinatorial graph, the configuration space is the set of unordered pairs of distinct vertices,

$S_N$3

and the dynamics can be formulated as a many-particle tight-binding model on the corresponding configuration graph (Harrison et al., 2011). In this setting, exotic quantum statistics emerge despite the graph being locally one-dimensional. Trees support only Bose statistics; circle graphs support a continuous anyon phase; graphs with multiple cycles allow multiple independent anyonic phases; and sufficiently complex nonplanar graphs such as $S_N$4 and $S_N$5 admit discrete-valued phases, typically $S_N$6 or $S_N$7, arising from torsion in the abelianized fundamental group or first homology (Harrison et al., 2011).

The higher-particle topological theory refines this picture. The universal presentation problem for graph configuration spaces is solved for wheel graphs $S_N$8, complete bipartite graphs $S_N$9, $\mathcal H_{MG}$0, and graphs with at most one essential vertex of degree $\mathcal H_{MG}$1, including trees (Maciążek et al., 2018). In many of these families, especially wheels and trees, the homology is torsion-free, so in sufficiently high rank all flat vector bundles are stably equivalent to the trivial bundle (Maciążek et al., 2018). This suggests that unlabeledness at the level of particle exchange can produce either a rich anyonic sector or a topologically rigid one, depending on the torsion structure of the graph configuration space rather than on the graph alone.

6. Labels, renaming invariance, and spectral identification

A common misconception is that unlabeled quantum graphs can always be built by quotienting the state space by graph isomorphism from the outset. The theory of quantum superpositions of graphs argues against this. The kinematic Hilbert space is taken to be

$\mathcal H_{MG}$2

where $\mathcal H_{MG}$3 is a named colored graph, and renamings act by

$\mathcal H_{MG}$4

Renaming-related basis states are orthogonal whenever $\mathcal H_{MG}$5, and node names are required for the correct alignment of degrees of freedom in graph superpositions; erasing them can lead to no-signaling violations and destroys the ability to define local observables consistently (Arrighi et al., 2020). The proposed remedy is not to quotient the state space, but to impose renaming invariance at the level of observables and dynamics. For local observables this takes the covariance form

$\mathcal H_{MG}$6

In this formulation, renaming invariance is the discrete analogue of diffeomorphism invariance (Arrighi et al., 2020).

Spectral theory shows that unlabeledness also does not trivialize graph identification. For equilateral connected quantum graphs with at most nine vertices, exhaustive computation found 364 isospectral sets, including thirteen isospectral triplets and one isospectral set of four; the constructions are directly relevant to the inverse problem for unlabeled quantum graphs because isospectrality is generated without reliance on vertex labeling (Pistol, 2021). At the same time, for leafless quantum graphs isospectral families for the standard Laplacian are finite, the minimum edge length is a spectral invariant, and the Bloch spectrum determines the Albanese torus, the block structure, the planarity, and a geometric dual of a planar graph; for planar 3-connected quantum graphs it completely determines the graph (Rueckriemen, 2011). Related work on graph quantization also states that the resulting spectra are independent of how the vertices are labeled or of graph isomorphism (Harrison, 2023). Taken together, these results show that unlabeledness and isomorphism-invariance do not eliminate inverse spectral ambiguity in general, but refined spectral data can recover full structure in specific metric classes.

7. Related graph-theoretic quantum representations

Several neighboring constructions connect unlabeled graph ideas to broader quantum-theoretic applications. A weighted-graph representation of quantum information interprets the density matrix of a quantum state as the normalized signless Laplacian of an associated weighted graph,

$\mathcal H_{MG}$7

with one-qubit states represented by a non-oriented weighted graph with two vertices and $\mathcal H_{MG}$8-qubit states by graphs on $\mathcal H_{MG}$9 vertices (Belhaj et al., 2016). The discussion explicitly notes that this interpretation paves the way for studying quantum information without explicit labels on graph vertices and relates the construction to graph isomorphism and symmetries in quantum systems (Belhaj et al., 2016).

Another line of work uses unlabeled bipartite graphs and MMP hypergraphs to represent Kochen–Specker setups, 3-dimensional spin systems, and lattices of Hilbert subspaces. Cubic bipartite graphs without labels correspond to classes of MMP hypergraphs, with atoms represented by white vertices, blocks by black vertices, and incidence by edges (Pavicic et al., 2010). This unlabeled formulation is used for exhaustive generation of structurally distinct quantum measurement scenarios and for checking quantum-logical constraints such as superposition and orthoarguesian equations (Pavicic et al., 2010).

These adjacent uses are not identical to operator-algebraic quantum graph theory, but they reinforce a common pattern. Unlabeledness is repeatedly used to isolate invariant structure from arbitrary naming, whether the objects are multigraph Hilbert states, automorphism game algebras, particle configuration spaces, weighted Laplacians, or hypergraph encodings of contextuality. The literature therefore supports no single canonical definition of unlabeled quantum graphs; instead it presents a technically diverse family of constructions centered on permutation invariance, isomorphism classes, and quantum symmetry.