spectral graph theory

This work addresses the challenge of efficiently computing leading eigenvectors in dynamic graphs, where frequent updates to the adjacency or Laplacian matrix render traditional eigendecomposition methods computationally prohibitive. To overcome this limitation, the authors propose a fast spectral embedding update framework based on Rayleigh-Ritz projection. By leveraging eigenvector perturbation analysis, the method constructs a low-dimensional approximate invariant subspace that preserves high approximation accuracy while substantially reducing computational and memory costs. Experimental results demonstrate that the proposed approach outperforms existing techniques in both the quality of leading eigenvector approximation and performance on downstream tasks—such as influential node identification and node clustering—offering a compelling balance between efficiency and accuracy.

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.

The computational complexity of graph isomorphism testing remains unresolved, particularly for highly symmetric graphs whose adjacency matrices possess repeated eigenvalues—causing ambiguity in the solution space for conventional methods. This paper introduces a novel continuous optimization framework: it reformulates the discrete matching problem via orthogonal and doubly stochastic relaxations, and—crucially—systematically characterizes how eigenvalue multiplicity governs the geometric structure of the feasible solution space. Building on this insight, we propose a subspace constraint strategy that effectively suppresses spurious solutions induced by symmetry. Our algorithm employs the Frank–Wolfe method, integrating spectral matrix analysis with iterative projection-based optimization. Extensive evaluation on highly symmetric benchmarks—including strongly regular graphs, complete graphs, and the Petersen graph—demonstrates substantial improvements in both efficiency and robustness of isomorphism detection.

Conventional diffusion models for graph generation suffer from O(n²) computational complexity in the node space, hindering scalability. Method: This paper proposes GGSD, the first model to jointly integrate graph Laplacian spectral decomposition with denoising diffusion probabilistic modeling, establishing a novel spectral-space diffusion paradigm. GGSD employs spectral truncation for efficient low-dimensional representation and introduces a linear-complexity, permutation-invariant Transformer architecture that supports node feature fusion. Crucially, it generates graph structures directly in the node space while achieving theoretical O(n) complexity. Contribution/Results: Extensive experiments demonstrate that GGSD significantly outperforms state-of-the-art methods on both synthetic and real-world graph datasets, achieving superior trade-offs among generation speed, structural fidelity, and scalability.

Multi-partite network spectral embeddings reside in high-dimensional spaces, yet their intrinsic node representations lie within group-specific low-dimensional subspaces—a geometric structure previously uncharacterized. Method: This paper introduces the first post-processing dimensionality reduction method with theoretical consistency guarantees to recover the intrinsic dimensionality of such embeddings. Grounded in a low-rank inhomogeneous random graph model, the method jointly leverages subspace estimation and matrix perturbation theory to provably achieve consistent subspace recovery; it further unifies and generalizes the bipartite spectral embedding framework. Results: Extensive experiments demonstrate that the proposed method significantly outperforms standard spectral embedding and conventional bipartite embedding approaches on clustering and visualization tasks, achieving both theoretical rigor and practical effectiveness.

This work addresses the tendency of Laplacian-constrained graphical models—such as the Laplacian-constrained Gaussian graphical model (LCGGM) and the Hüsler–Reiss model—to yield overly dense graphs during structure learning, which compromises interpretability and scalability. The paper introduces, for the first time, spectral graph sparsification as a post-processing step: without requiring additional hyperparameter tuning, it replaces the original Laplacian estimate with a spectrally approximated sparse Laplacian and refits the model. This approach effectively enhances both the sparsity and accuracy of the estimated graph structure. Integrating spectral graph theory, Laplacian-constrained Gaussian graphical models, extreme-value graphical models, and graph sparsification techniques, the method demonstrates superior performance on Erdős–Rényi and stochastic block model simulations and validates its practical utility on real-world data.

This work addresses the high computational cost of computing multiple nonlinear eigenvectors in nonlinear spectral clustering by proposing a direct multiway spectral clustering algorithm based on the p-norm (with p ∈ (1,2]). The algorithm is implemented for the first time within the C++ GraphBLAS framework, unifying its core operations into sparse linear algebraic expressions. By integrating shared-memory parallelism and p-norm-specific optimizations, the approach achieves substantial gains in computational efficiency. Experimental results on a large-scale graph with 8 million nodes and 48 million edges demonstrate excellent strong scaling performance, while the clustering quality surpasses that of existing methods in terms of balanced graph cut metrics.

This study addresses the spectral determination of graphs and the edge- and distance-transitivity of generalized Hamming graphs and their complements. By integrating tools from spectral graph theory, group actions, and association schemes—including the Cauchy interlacing theorem, Schur complements, and Cayley graph constructions—the work introduces a novel family of “pyramid graphs” and establishes their adjacency-spectral uniqueness. It fully characterizes the conditions under which generalized Hamming graphs and their complements are edge-transitive or distance-transitive, and derives closed-form expressions for their Lovász ϑ-functions. The results yield new spectral characterizations of complete bipartite graphs, Turán graphs, and strongly regular graphs, confirm the spectral uniqueness of pyramid graphs, and systematically uncover the intrinsic connections between the symmetry and spectral properties of generalized Hamming graphs.

Existing nonparametric statistical testing methods based on surrogate data are primarily designed for undirected graphs and are ill-suited for directed graph structures. This work extends such approaches to the directed graph setting for the first time by defining wide-sense stationary signals through the eigendecomposition of graph shift operators and constructing a surrogate signal generation framework that preserves the covariance structure. Evaluated on real-world data, the proposed method significantly outperforms conventional undirected-graph approaches and naive permutation strategies, offering enhanced statistical power while maintaining test validity and practical feasibility.

This work addresses the limitation of traditional graph Laplacians in capturing memory effects in complex network diffusion, particularly their inability to model the influence of backtracking paths. The authors propose a novel class of walk-based Laplacian operators that flexibly incorporate memory by either excluding or reweighting backtracking walks, yielding three variants: fully traversing, non-backtracking, and continuously interpolated. This framework uniquely integrates non-backtracking and backtracking-weighted mechanisms into the Laplacian formalism, enabling controllable modeling of path memory while preserving certain classical spectral properties. Leveraging Krylov subspace methods and GPU acceleration, the approach scales efficiently to large-scale networks. Experiments on multiple real-world datasets demonstrate its superior expressiveness and computational efficiency, significantly enhancing the modeling of diffusion dynamics in complex networks.