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arxiv: 2606.29391 · v1 · pith:EKJTVWNJ · submitted 2026-06-28 · math.CO

Minimal Isometric Embeddings of Graphs into Abelian Groups: Theory, Algorithms, and Applications to Signal Processing over Networks

Fokam Souop Rigobert , Bitjoka Laurent This is my paper

Reviewed by Pith2026-06-30 02:20 UTCgrok-4.3pith:EKJTVWNJopen to challenge →

classification math.CO
keywords isometric embeddingCayley graphabelian groupgraph signal processingmetric parallelismquotient labelingharmonic analysisnetwork Fourier transform
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The pith

Every connected graph on n vertices isometrically embeds into a Cayley graph of (Z_2)^k with k at most n-1.

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

The paper develops a universal algorithm to embed any connected graph isometrically into the Cayley graph of an abelian group, specifically (Z_2)^k with dimension at most n-1. It defines edge relations phi, Phi, and Psi that detect metric parallelism, applies a transitive prune to obtain same-generator partitions, and uses the Cocycle/Quotient Labeling Theorem to assign consistent vertex labels in the group. The labeling succeeds except for shortcuts, which a repair loop corrects to achieve isometry. This construction grounds Fourier analysis, convolution, and wavelets directly on the embedded graph signals rather than through matrix approximations. A sympathetic reader would care because the algebraic host supplies exact translation-modulation duality and Plancherel identities for irregular network data.

Core claim

The central claim is that any connected graph G admits an isometric embedding into a Cayley graph of (Z_2)^k with k <= n-1; the proof proceeds by introducing edge relations that capture metric parallelism, pruning them transitively into candidate partitions, and applying the Cocycle/Quotient Labeling Theorem to obtain a GF(2) quotient labeling of dimension k = t - rank(A) that fails only by shortcuts, which are repaired via an isometric spanning-tree embedding, with the result verified exhaustively on all 995 connected graphs of at most seven vertices and extended to products of cyclic groups via the Smith Normal Form.

What carries the argument

The Cocycle/Quotient Labeling Theorem, which turns any edge partition into a most-generic consistent vertex labeling as a GF(2) quotient whose dimension is t minus the rank of the cycle-class parity matrix and fails only by shortcuts.

If this is right

  • The minimal embedding dimension satisfies k >= max(diam(G), ceil(log2 n)).
  • Stars K_{1,q} admit embeddings with dimension ceil(log2 q) + 1, exponentially below the naive bound.
  • Odd cycles require dimension exactly n-1.
  • The same quotient machinery generalizes via Smith Normal Form to embeddings into arbitrary finite abelian groups.
  • The embeddings preserve convolution theorems, translation-modulation duality, and Plancherel identities for graph signals.

Where Pith is reading between the lines

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

  • The algorithm could be run on larger random graphs to test whether k <= n-1 remains tight or admits tighter universal bounds.
  • Minimal-dimension search might be implemented by enumerating pruned partitions and selecting the smallest successful labeling.
  • The framework suggests that many matrix-based graph Fourier transforms are special cases of the group Fourier transform on the host abelian group.
  • Bipartite graphs may recover known hypercube or partial-cube embeddings as special cases of the same construction.

Load-bearing premise

The transitive prune on the edge relations always produces partitions that let the labeling theorem yield consistent labelings failing only by shortcuts that the repair loop can correct to isometry.

What would settle it

A single connected graph on eight or more vertices for which the algorithm produces no isometric embedding into any (Z_2)^k Cayley graph with k <= n-1.

Figures

Figures reproduced from arXiv: 2606.29391 by Bitjoka Laurent, Fokam Souop Rigobert.

Figure 1
Figure 1. Figure 1: Illustration of the embedding concept: An irregular graph [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of a partial cube embedding (red) into a host hypercube [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The gap between Classical Signal Processing (rigid algebra) and Graph Signal [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 1.1
Figure 1.1. Figure 1.1: Left: A Cartesian product grid graph. Right: The Petersen graph, featuring [PITH_FULL_IMAGE:figures/full_fig_p035_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: The cyclic group C6 = ⟨r⟩ acting on the 6-cycle graph C6 by rotation. The six rim edges form the cycle itself; the generator r rotates the hexagon by 60◦ , sending each vertex vi to its neighbour vi+1 (indices mod 6). Because r carries edges to edges, it is a graph automorphism. These matrices satisfy: • A encodes the immediate connectivity. • L acts as a difference operator: (Lf)i = P j∼i (f(i) − f(j)).… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Visualization of the periodic extension (wraparound) mapping a 1D finite [PITH_FULL_IMAGE:figures/full_fig_p039_1_3.png] view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: Schematic of a 2D grid with periodic boundary conditions (wraparound), [PITH_FULL_IMAGE:figures/full_fig_p040_1_4.png] view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: Schematic of a Two-Channel Filter Bank for Wavelet Transform on Groups, [PITH_FULL_IMAGE:figures/full_fig_p040_1_5.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Non-transitivity of φ in K2,3: a φ b and b φ c, yet a ̸φ c since a and c are incident at u1. The relation captures local parallelisms that may conflict globally; the transitive prune of Section 2.4 resolves the conflicts [PITH_FULL_IMAGE:figures/full_fig_p050_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: The five φ-equivalence classes F1, . . . , F5 of the Petersen graph, each of car￾dinality 3. Since φ is already an equivalence relation here, no transitive prune is needed; the five classes directly determine the embedding into Z 4 2 via the Quotient Labeling The￾orem [PITH_FULL_IMAGE:figures/full_fig_p050_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: The hypercube skeleton of the Petersen graph: four of the five [PITH_FULL_IMAGE:figures/full_fig_p054_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: The Petersen graph (red) as an isometric subgraph of the Clebsch graph [PITH_FULL_IMAGE:figures/full_fig_p057_2_4.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: The nine structured φ −-classes of the Pappus graph (t = 9, three edges per class). The cycle–class parity matrix has rank 2, so the quotient embedding has k = 7: host order 128, verified isometric, with both composite generators of weight 5. 2.7 The φ-Quotient Algorithm 2.7.1 Isometry Check and Shortcut Characterization Definition 2.30 (Correct isometry check). An embedding λ: V (G) → Z k 2 with generat… view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: The optimal embedding of the star K1,4 into Cay(Z 3 2 , {001, 010, 100, 111}): center 7→ 000, leaves on a sum-free generator set. All leaf pairs are at Cayley distance exactly 2 (011, 101, 110 ∈/ S). Since idim(K1,4) = 4, composite generators strictly beat the hypercube paradigm even on trees. an even closed walk, and every closed walk of even length contains an even cycle unless it backtracks entirely; … view at source ↗
Figure 2.7
Figure 2.7. Figure 2.7: The interval framework on C7. Generators along any geodesic arc (here W = {s0, s1, s2}, |W| = diam = 3) are linearly independent (Lemma 2.34); any dependency x /∈ {0, 1} among all seven generators produces, on some arc, a generator subset shorter than the cycle distance — a shortcut. Hence the seven generators have rank 6 and kmin(C7) = 6. v0 0000 v1 1111 v2 1011 v3 1001 v4 1000 1111 1000 0100 0010 0001 … view at source ↗
Figure 2.8
Figure 2.8. Figure 2.8: The φ-quotient embedding of C5: five singleton classes, one cycle relation, k = 5 − 1 = 4, with the four basis generators and the composite e1+e2+e3+e4 on the closing edge. By Theorem 2.40 this dimension is minimal, so the algorithm is optimal on C5 [PITH_FULL_IMAGE:figures/full_fig_p064_2_8.png] view at source ↗
Figure 2.9
Figure 2.9. Figure 2.9: Distribution of the final embedding dimension [PITH_FULL_IMAGE:figures/full_fig_p066_2_9.png] view at source ↗
Figure 2.10
Figure 2.10. Figure 2.10: Repair-loop rounds to convergence over all 995 connected graphs with [PITH_FULL_IMAGE:figures/full_fig_p067_2_10.png] view at source ↗
Figure 2.11
Figure 2.11. Figure 2.11: Algorithm dimension vs. exact minimum over all 30 connected graphs with [PITH_FULL_IMAGE:figures/full_fig_p067_2_11.png] view at source ↗
Figure 2.13
Figure 2.13. Figure 2.13: Petersen graph embedding into Cay(Z 4 2 , S) (host order 16): skeleton edges in red, each vertex labeled with its 4-bit coordinate, the five φ −-classes in distinct colors [PITH_FULL_IMAGE:figures/full_fig_p069_2_13.png] view at source ↗
Figure 2.12
Figure 2.12. Figure 2.12: The four GSP benchmark domains reused in Part II: ring, image grid, [PITH_FULL_IMAGE:figures/full_fig_p070_2_12.png] view at source ↗
Figure 2.14
Figure 2.14. Figure 2.14: Diamond graph (K4 minus an edge) embedded into Cay(Z 3 2 , S) of order 8: two φ-pairs define two cuts, the remaining edges are singleton classes. 00 10 01 11 Complete K4 (n=4, m=6, p=2, host=2^2=4, iso= ) Skeleton Non-skeleton 0 1 2 3 -classes (3 classes) C1 (2 edges) C2 (2 edges) C3 (2 edges) [PITH_FULL_IMAGE:figures/full_fig_p071_2_14.png] view at source ↗
Figure 2.15
Figure 2.15. Figure 2.15: K4 embedded into Cay(Z 2 2 , S) of order 4 with S = Z 2 2 \ {0}: ε = 1, the graph fills its host (Proposition 2.36(ii) with t = 2). 000 101 100 111 011 010 Cycle C6 (n=6, m=6, p=3, host=2^3=8, iso= ) Skeleton Non-skeleton 0 2 1 3 4 5 -classes (3 classes) C1 (2 edges) C2 (2 edges) C3 (2 edges) [PITH_FULL_IMAGE:figures/full_fig_p071_2_15.png] view at source ↗
Figure 2.16
Figure 2.16. Figure 2.16: C6 embedded into Q3: the three antipodal φ-classes of Theorem 2.12(v) are the three coordinates; k = 3 = diam, optimal [PITH_FULL_IMAGE:figures/full_fig_p071_2_16.png] view at source ↗
Figure 2.17
Figure 2.17. Figure 2.17: 3×3 grid embedded into Q4 (ε = 0.562): two horizontal and two vertical cut classes realize the product structure P3 □ P3 ⊆ Q2 □ Q2. 00000 10000 10010 11010 01010 01000 01001 01101 01111 01110 00110 10110 10111 11111 11011 11001 10001 00101 10101 00100 Desargues (n=20, m=30, p=5, host=2^5=32, iso= ) Skeleton Non-skeleton 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 17 19 -classes (5 classes) C1 (6 edges) … view at source ↗
Figure 2.18
Figure 2.18. Figure 2.18: Desargues graph embedded into Cay(Z 5 2 , S) of order 32: five perfect￾matching classes, no cycle relations (ρ = 0), k = 5; the graph is a partial cube and the embedding attains ε = 0.625. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 25 24 26 27 28 29 30 31 32 33 Karate Club (n=34, m=78, p=17, host=2^17=131072, iso=?) Skeleton Non-skeleton 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 … view at source ↗
Figure 2.19
Figure 2.19. Figure 2.19: Zachary Karate Club [86] (n = 34, m = 78): a triangle-dense real-world graph. The φ − classes shatter under repair and the dimension cap returns the naive embedding k = 33; the graph marks the practical frontier of the purely binary method and the entry point of the cyclic factors of Chapter 3. 2.11 Discussion Against the naive method. The quotient algorithm achieves exponential host reduc￾tions on stru… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The corrected class constraint. In K3 = Cay(Z3, {1, 2}) one generator carries all three edges as a directed cycle. Classes are partial permutations (directed paths and cycles), not matchings; matchings are the involution case 2g = 0 [PITH_FULL_IMAGE:figures/full_fig_p075_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Host orders: abelian (Chapter 3) vs. binary (Chapter 2). Odd cycles and [PITH_FULL_IMAGE:figures/full_fig_p082_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: CL5 recognized as the abelian Cayley graph Cay(Z5 × Z2, {(±1, 0),(0, 1)}): the Ψ chain merge assembles the ten ring edges into one directed class, the ring relation 5g = 0 emerges in the SNF, and the host attains the lower bound. initializer portfolio ( 4-cycle UF, chain-merge, cycle/path/complete) oriented partition (partial permutations) + signed matrix A Smith Normal Form univ = t/rowlat(A) di × f sub… view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: The Chapter 3 pipeline. The Φ/Ψ machinery serves as the initializer portfolio [PITH_FULL_IMAGE:figures/full_fig_p083_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: The binary-ground phenomenon. Over all connected graphs on at most seven [PITH_FULL_IMAGE:figures/full_fig_p085_3_5.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Circular ladder CL5 ,→ Cay(Z5 × Z2, {(±1, 0),(0, 1)}), host order 10 = n, provably minimal — contrast the binary host 32. (0,0) (1,0) (2,0) (3,0) (4,0) (0,1) (1,1) (2,1) (3,1) (4,1) Group-theoretic routing on CL5 in 5× 2: = (2, 1) decomposes greedily as (1, 0) + (1, 0) + (0, 1) no routing tables, only coordinate arithmetic [PITH_FULL_IMAGE:figures/full_fig_p087_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Coordinate routing on the same embedding (used in Chapter 4): the desti [PITH_FULL_IMAGE:figures/full_fig_p087_3_7.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Table-free routing on CL5 hosted in Z5 ×Z2. To route from (0, 0) to (2, 1) the offset δ = (2, 1) is reduced by the word (1, 0) + (1, 0) + (0, 1); each node makes a purely local, table-free coordinate decision [PITH_FULL_IMAGE:figures/full_fig_p090_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Coset task assignment for the 3×3 grid in its host: vertices colored by the four cosets of an index-4 subgroup, giving a balanced partition with communication along quotient edges. Proposition 4.11 (Collective primitives). On H = Cay(Γ, S): broadcast runs in O(diam(H)) along a BFS tree; all-reduce of f is P g f(g) = N ˆf(1), the FFT at the trivial character; and any permutation routes in O(diam(H)) gener… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Documented HV transmission backbone, plotted on Cameroon’s actual [PITH_FULL_IMAGE:figures/full_fig_p098_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: The same backbone with the recommended augmentation (Plan B, capacity [PITH_FULL_IMAGE:figures/full_fig_p099_4_4.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Two Fourier bases on the Petersen graph. Left: a Laplacian eigenvector from a [PITH_FULL_IMAGE:figures/full_fig_p104_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Denoising gains from Table 5.1, visualized. The structured benchmark graphs [PITH_FULL_IMAGE:figures/full_fig_p110_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Genuine translation on C16 = Cay(Z16, {±1}): the operator T3 rigidly shifts a vertex signal by three positions — a group action with TaTb = Ta+b, impossible in spectral GSP. 0.0 0.1 0.2 0.3 0.4 0.5 character frequency norm 0.0 0.1 0.2 0.3 0.4 0.5 spectral energy |s(k)| 2 Group Fourier spectrum of a smooth signal on C16: energy concentrates at low frequency the convolution theorem and a true low/high-freq… view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: GE-GFT spectrum of a smooth (low-Laplacian-mode) signal on [PITH_FULL_IMAGE:figures/full_fig_p110_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Translation isometry, computed. Left: the spectral-GSP generalized trans [PITH_FULL_IMAGE:figures/full_fig_p111_5_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Basis uniqueness, computed. Left: the K3,3 Laplacian spectrum has a multiplicity-four eigenspace (red), within which the spectral GFT basis is ambiguous up to a 4 × 4 unitary rotation. Right: the canonical character basis of the group host Z2 × Z4 has no such ambiguity. 5.6.3 Denoising: Group DFT vs. Laplacian Filtering We replicate the denoising setting of Shuman et al.’s Tikhonov example [56, Ex. 2], w… view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Denoising on the 8 × 8 grid, computed. The group 2-D DFT low-pass (third panel, +4.3 dB) recovers the smooth structure; Laplacian Tikhonov filtering with an equal smoothness weight (fourth panel, +0.5 dB) over-smooths. On the ring the two methods are identical. 5.6.4 Wavelets: Spectral Graph Wavelets vs. the Group Tight Frame The spectral graph wavelet transform (SGWT) of Hammond et al. [31] represents a… view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: The wavelet filter bank on the host (here [PITH_FULL_IMAGE:figures/full_fig_p119_6_1.png] view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Group wavelet atoms on C32 centered at vertex 0, at three scales. Fine-scale atoms (left) are sharply localized; coarse-scale atoms (right) spread over the ring while remaining centered. The canonical, translation-covariant localized analyzing functions that spectral graph wavelets cannot provide [PITH_FULL_IMAGE:figures/full_fig_p120_6_2.png] view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: Multiresolution decomposition on C64 of a signal that is low-frequency on the first half and high-frequency on the second. The approximation captures the global trend; successive detail bands isolate the high-frequency burst and localize it to the correct half of the ring — joint vertex–frequency analysis on a graph, computed via the host FFT [PITH_FULL_IMAGE:figures/full_fig_p121_6_3.png] view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: Two-dimensional group wavelet atoms on the 8 [PITH_FULL_IMAGE:figures/full_fig_p122_6_4.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: Graph signal denoising on C64. Left: clean, noisy (4.3 dB) and Wiener-filtered (16.7 dB) signals. Right: the Wiener spectral shrinkage, attenuating high-frequency (noise-dominated) coefficients while preserving the low-frequency signal peaks. All values computed on the reference host. 7.2 Compression by Coefficient Thresholding Smooth graph signals have sparse spectra, so retaining the largest transform … view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: Rate–distortion for the 8×8 grid image on the host torus Z 2 14 (left, PSNR vs. retained coefficients, dashed line at 35%) and the reconstruction at 20% of coefficients (right). Computed on the reference host. 7.3 Anomaly Detection by High-Pass Filtering A localized anomaly injected into a smooth signal is, by the uncertainty principle (The￾orem 5.11), spread across the spectrum; its high-frequency footp… view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: Anomaly detection on C12(1, 2). Left: a smooth signal with an injected spike at vertex 5. Right: the high-pass residual, peaking at the correct vertex. Computed on the reference host. 7.4 Distributed Consensus on the Host Consensus dynamics ˙x = −Lx converge at a rate set by the algebraic connectivity λ2. On a Cayley host the Laplacian diagonalizes in the character basis, giving λ2 in closed form. Theore… view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: Real AWS EC2 CPU-utilization (instance 5f5533, 14 days at 5-min resolu￾tion) arranged on the host torus Z14 × Z288. Left: the signal as a day×time-of-day map; a regime change near day 11 and two short spikes are visible. Right: the mean intraday profile, i.e. the low-frequency content the host transform isolates. 7.5.2 Compression, denoising, and anomaly detection (computed) Compression (Theorem 7.4). Re… view at source ↗
Figure 7.5
Figure 7.5. Figure 7.5: Left: rate–distortion for best K-term compression of the AWS trace on the host (2-D DFT). Right: the unsupervised high-pass residual over the 14-day trace; the two NAB ground-truth anomaly windows are shaded, and the two largest residual peaks in the entire series fall inside them. All values computed on the reference host. 7.5.3 Why this matters for industry, and a scope note Cloud and data-center opera… view at source ↗
read the original abstract

This dissertation develops a framework for embedding arbitrary connected graphs isometrically into Cayley graphs of abelian groups, with applications to harmonic analysis on networks. It addresses representing irregular graph-structured data within highly symmetric algebraic hosts, on which classical Fourier theory applies verbatim rather than by analogy. The theoretical core is twofold. First, we introduce edge relations phi, Phi, and Psi that detect metric parallelism, a strict generalization of the Djokovic-Winkler relation beyond bipartite and partial-cube structures, with a transitive prune operation converting them into candidate same-generator edge partitions. Second, we prove the Cocycle/Quotient Labeling Theorem: any edge partition induces a most-generic consistent vertex labeling as a GF(2) quotient of dimension k = t - rank(A), where A is the cycle-class parity matrix; the labeling can fail only by shortcuts, never by stretching. With a shortcut-repair loop terminating in the isometric spanning-tree embedding, this gives a universal algorithm: every connected graph G embeds isometrically into a Cayley graph of (Z_2)^k with k <= n-1, verified exhaustively on all 995 connected graphs of at most seven vertices. A bounds theory follows: k >= max(diam(G), ceil(log2 n)); stars satisfy k_min(K_{1,q}) = ceil(log2 q) + 1, exponentially below the naive dimension; odd cycles require k = n-1. We then generalize the quotient machinery from GF(2) to Z via the Smith Normal Form, giving embeddings into products of cyclic groups. The primary application is harmonic analysis: these embeddings ground Fourier analysis, convolution, and wavelet transforms on graph signals, preserving translation-modulation duality, convolution theorems, and Plancherel identities that matrix-based graph signal processing lacks. We name this framework Group-Embedding-based Graph Signal Processing (GE-GSP).

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.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a framework for isometric embeddings of arbitrary connected graphs into Cayley graphs of abelian groups, centered on metric parallelism relations (phi, Phi, Psi) with a transitive prune, the Cocycle/Quotient Labeling Theorem producing GF(2) labelings from edge partitions (failing only by shortcuts), and a shortcut-repair loop yielding embeddings into (Z_2)^k with k ≤ n-1. It includes bounds (e.g., stars satisfy k_min = ceil(log2 q) + 1; odd cycles require k = n-1), a generalization to Z via Smith Normal Form, exhaustive verification on all 995 connected graphs with n ≤ 7, and applications to Group-Embedding-based Graph Signal Processing (GE-GSP) preserving Fourier properties exactly.

Significance. If the universal embedding result and repair procedure hold, the work supplies a rigorous algebraic host for exact harmonic analysis, convolution, and wavelets on irregular graphs, addressing a core limitation of matrix-based GSP. The parameter-free nature of the construction, the explicit bounds for stars and cycles, and the exhaustive verification on all small graphs are notable strengths that support falsifiability and reproducibility.

major comments (1)
  1. [Cocycle/Quotient Labeling Theorem] Cocycle/Quotient Labeling Theorem: the claim that the shortcut-repair loop always terminates in an isometric spanning-tree embedding (yielding the universal k ≤ n-1 result) is load-bearing; the manuscript must supply an explicit termination argument that applies beyond the n ≤ 7 exhaustive check, as the theorem statement indicates failure occurs only by shortcuts but does not detail why the loop cannot cycle or produce non-isometric results on larger graphs.
minor comments (2)
  1. The count of 995 connected graphs with n ≤ 7 should be verified against standard enumerations (e.g., OEIS A001349) and stated with the exact enumeration source.
  2. Notation for the cycle-class parity matrix A and the rank computation k = t - rank(A) should be defined explicitly at first use in the theorem statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the constructive major comment. We address it point by point below.

read point-by-point responses

  1. Referee: [Cocycle/Quotient Labeling Theorem] Cocycle/Quotient Labeling Theorem: the claim that the shortcut-repair loop always terminates in an isometric spanning-tree embedding (yielding the universal k ≤ n-1 result) is load-bearing; the manuscript must supply an explicit termination argument that applies beyond the n ≤ 7 exhaustive check, as the theorem statement indicates failure occurs only by shortcuts but does not detail why the loop cannot cycle or produce non-isometric results on larger graphs.

    Authors: We agree that an explicit termination argument is required to support the universal result beyond the n ≤ 7 verification. The current text states that the loop terminates in an isometric spanning-tree embedding but relies on the theorem's shortcut-only failure mode without a general proof against cycling or non-isometry. In revision we will add a formal argument: each repair step eliminates at least one shortcut (by adjusting the GF(2) labels on the affected partition) while preserving distances on all non-shortcut edges and never increasing the quotient dimension k; a potential function equal to the number of shortcut edges therefore strictly decreases, guaranteeing termination after finitely many steps with no cycles possible. The resulting labeling is isometric by the theorem's guarantee that the only possible inconsistencies are shortcuts. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation chain rests on the stated Cocycle/Quotient Labeling Theorem (which produces GF(2) labelings from edge partitions after transitive prune, failing only by shortcuts) together with a repair loop that reaches an isometric spanning-tree embedding; the universal bound k ≤ n-1 follows directly from this construction. Exhaustive verification on all 995 connected graphs with n ≤ 7 supplies independent confirmation rather than a fit to the same data. No self-citations, fitted parameters renamed as predictions, or self-definitional reductions appear in the provided chain; the result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper relies primarily on standard mathematical axioms and introduces new concepts for the embedding process. No numerical parameters are fitted; the dimension k is bounded theoretically.

axioms (2)
  • standard math Basic axioms of graph theory and abelian group theory
    The framework builds on standard definitions of graphs, distances, and group operations.
  • domain assumption Existence of edge partitions from the relations that allow consistent labeling
    This is central to the Cocycle/Quotient Labeling Theorem as described.
invented entities (2)
  • Metric parallelism edge relations (phi, Phi, Psi) no independent evidence
    purpose: Detecting parallel edges in the graph metric to form partitions
    Newly proposed in the paper to extend beyond bipartite cases.
  • Cocycle/Quotient labeling no independent evidence
    purpose: Assigning group elements to vertices based on edge partitions
    Core mechanism for the embedding.

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Reference graph

Works this paper leans on

195 extracted references · 4 canonical work pages

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