group action enumeration

This study addresses the efficient enumeration of connected graphs whose automorphism groups act with exactly two orbits. To this end, the authors propose a novel methodology that integrates Goursat’s lemma to construct candidate groups, performs graph enumeration under automorphism group constraints, and incorporates group-theoretic pruning to enhance computational efficiency. This approach yields the first complete enumeration of all connected two-orbit graphs on up to 27 vertices, resulting in a total of 10,094,721 such graphs. The method substantially surpasses the limitations of traditional brute-force enumeration techniques, dramatically expanding the scale of instances that can be feasibly solved within this class of symmetry-constrained graph enumeration problems.

This work addresses four #P-complete graph counting problems in computational chemistry—Kekulé structure enumeration, Hosoya index, Merrifield–Simmons index, and matching/independent set entropy—by introducing the first fixed-parameter tractable (FPT) framework parameterized by treewidth (tw) and pathwidth (pw). We design a unified dynamic programming algorithm for these chemical topological indices, achieving time complexity $O^*(2^{O( ext{tw})})$, substantially improving upon existing exponential-time baselines. Theoretical analysis and empirical evaluation on the full PubChem dataset (>100 million compounds) demonstrate that >99.9% of real chemical graphs satisfy $ ext{tw} leq 6$, validating the small-treewidth hypothesis. Our implementation scales efficiently to large-scale chemical graphs, delivering speedups of several orders of magnitude over naive enumeration. This work bridges parameterized algorithms and cheminformatics, establishing a new paradigm for exact computation of #P-hard chemical graph metrics.

Canonical labeling of highly symmetric random circulant graphs—including their directed variants—remains challenging, as conventional combinatorial approaches such as color refinement fail on vertex-transitive graphs due to symmetry-induced indistinguishability. Method: This paper introduces a novel hybrid framework integrating color refinement with vertex individualization. Its core innovation lies in deriving a unique canonical label solely from the counts of walks of all lengths from each vertex to the individualized vertex. The method unifies Tinhofer’s canonicalization procedure, the 2-dimensional Weisfeiler–Leman algorithm, and walk-counting analysis to overcome refinement stagnation caused by automorphic symmetry. Results: Experiments demonstrate efficient canonical labeling for almost all random circulant (and circulant directed) graphs, along with construction of their canonical Cayley representations. This significantly advances the theoretical frontiers of graph isomorphism testing and encoding of symmetric graphs.

This paper investigates the computational complexity of counting weighted Eulerian orientations (#EO) within the Holant framework—a long-standing open problem. Methodologically, it integrates Holant theory, algebraic invariant analysis of signatures, graph gadget construction, and closure property proofs. The contributions include: (i) the first complete complexity classification for three signature classes—binary/quaternary, pure, and rebalanced; (ii) the first polynomial-time algorithm capable of handling non-pure signatures (e.g., $f_{40}$), overcoming prior restrictions to pure signatures; and (iii) three dichotomy theorems that rigorously delineate tractable versus #P-hard cases. Empirical validation confirms the algorithm’s correctness on canonical non-pure instances such as $f_{40}$. The complexity of $f_{56}$ remains unresolved.

This work investigates the discriminative power of spectral angles—the angles between eigenvectors of the adjacency matrix and standard basis vectors—for graph isomorphism testing. Methodologically, it establishes a purely combinatorial characterization of spectral angles at the level of walk counts, enabling a precise comparison with the Weisfeiler–Leman (WL) hierarchy. The contributions are threefold: (i) it proves that spectral angles are strictly equivalent in expressive power to the 2-dimensional WL algorithm (2-WL), yet strictly weaker than 3-WL—thereby fully resolving an open problem posed by Fürer regarding this invariant; (ii) it uncovers intrinsic connections between spectral angles, generalized spectra, and principal spectra; and (iii) it demonstrates that “almost all graphs are uniquely determined by their spectrum together with spectral angles”, yielding significant progress toward the long-standing conjecture on spectral uniqueness of graphs.

This work investigates the algebraic–combinatorial mechanisms underlying graph isomorphism discrimination and introduces the framework of “separating modules,” a polynomial vector space grounded in the representation theory of symmetric groups. By employing complexity measures such as support size, symmetric circuit size, and multiplicity, it establishes equivalences with subgraph counting (support size \(k\) corresponds to order \(O(k)\)) and the Weisfeiler–Leman algorithm (circuit size \(n^{\Theta(k)}\) corresponds to \(\Theta(k)\)-WL). The central contribution provides the first intrinsic characterization of multiplicity separation: two graphs are distinguishable if and only if their automorphism groups have distinct cycle indices. Furthermore, the paper demonstrates that the multiplicity barrier is strictly stronger than the occurrence barrier and connects invariant polynomials to the graph reconstruction conjecture and finite-type invariants.

This work investigates the parameterized counting complexity of $k$-vertex induced subgraphs satisfying a fixed graph property $\Phi$, with a focus on symmetry conditions dictated by the structure of their automorphism groups. By initiating from the $k$-clique problem and employing a refined parameterized reduction based on a “clique gadget” construction, the study establishes—for the first time—that counting $k$-vertex induced subgraphs whose automorphism group is exactly a given finite group $Q$ is $\#\mathbf{W}[1]$-hard for any finite group $Q$. This result not only confirms the $\#\mathbf{W}[1]$-hardness in the case of trivial automorphism groups but also generalizes it to arbitrary finite groups, thereby overcoming limitations inherent in existing Fourier-analytic approaches and resolving a long-standing open problem in this direction.

This work addresses the problem of efficiently ranking and unranking permutations in Catalan classes that avoid patterns of length three, under lexicographic or co-lexicographic order. For these permutation classes—enumerated by the Catalan numbers—we present the first systematic linear-time algorithms, thereby filling a notable gap in combinatorial generation. Our approach leverages a structural analysis of the underlying combinatorial objects, combined with recursive decomposition and lexicographic traversal techniques, to enable fast encoding and decoding of any such permutation. The resulting framework not only offers theoretical completeness but also establishes a foundation for practical applications requiring efficient enumeration and indexing within these Catalan families.

This work develops a novel algorithmic information theory within the framework of symmetric groups to characterize string complexity induced by symmetries. By introducing symmetry groups generated by computable bijections, the authors define a “symmetric prior” and, under the fix-retractable condition, prove it constitutes a universal lower-semicomputable semimeasure, thereby establishing a geometric coding theorem. The central innovation lies in the first unified integration of algorithmic information theory with group theory, proposing a new paradigm for complexity measures grounded in symmetry. Furthermore, the study reveals a structural correspondence between subgroups and sets of binary strings via a Galois connection. This theoretical foundation advances computational algorithmic statistics (CAS) and opens new avenues for analyzing structured data through algebraic and informational lenses.

Traditional graph representations face significant challenges in graph isomorphism testing and symmetry-aware visualization due to high computational complexity and low efficiency. This work proposes “graph linear notation”—a complete graph invariant derived from canonical form algorithms—and establishes it, for the first time, as an equivalent definition for finite graphs. This representation not only substantially simplifies graph isomorphism comparison and symmetry-aware visualization but also naturally accommodates the extension and application of classical graph-theoretic concepts, such as coloring and paths, within its framework. By unifying these capabilities, the proposed notation offers a highly efficient and coherent new paradigm for structural graph analysis.