Search Results (65)

Graph Neural Networks (GNNs) face significant challenges in node classification across diverse graph structures. Traditional message passing mechanisms often fail to adaptively weight node relationships, thereby limiting performance in both homophilic and heterophilic graph settings. We propose the Eigenvector Distance-Modulated Graph Neural Network (EDM-GNN), which enhances message passing by incorporating spectral information from the graph’s eigenvectors. Our method introduces a novel weighting scheme that modulates information flow based on a combined similarity measure. This measure balances feature-based similarity with structural similarity derived from eigenvector distances. This approach creates a more discriminative aggregation process that adapts to the underlying graph topology. It does not require prior knowledge of homophily characteristics. We implement a hierarchical neighborhood aggregation framework that utilizes these spectral weights across multiple powers of the adjacency matrix. Experimental results on benchmark datasets demonstrate that EDM-GNN achieves competitive performance with state-of-the-art methods across both homophilic and heterophilic settings. Our approach provides a unified solution for node classification problems with strong theoretical foundations in spectral graph theory and significant empirical improvements in classification accuracy. Full article

The internal structure of hyperedges has become central to understanding collective dynamics in hypernetworks. This study investigates the impact of hyperedge overlap on network synchronization when hyperedge structures are explicitly considered. We propose a modified hyper-adjacency matrix that captures the internal organization of the hyperedges while preserving the higher-order properties. Using this framework, we examine how non-complete connections within hyperedges influence synchronization as the overlap increases. Our findings reveal clear differences from fully connected hyperedge models. Furthermore, spectral graph theory and numerical simulations confirm that the structural variations induced by overlaps significantly regulate global synchronization. This work extends the theoretical framework of hypernetwork synchronization and highlights the critical role of hyperedge overlaps in shaping the internal hyperedge structure. Full article

Spectral-domain Optical Coherence Tomography (SD-OCT) is a critical tool in ophthalmology, providing high-resolution cross-sectional images of the retina. Accurate segmentation of sub-retinal layers is essential for diagnosing and monitoring retinal diseases. While manual segmentation by clinicians is the gold standard, it is subjective, time-intensive, and impractical for large-scale use. This study introduces an automated segmentation algorithm based on graph theory, utilizing a shortest-path graph-search technique to delineate seven intra-retinal boundaries. The algorithm incorporates a region of interest (ROI) selection to enhance efficiency, achieving a mean computation time of 0.93 s on standard systems suitable for real-time clinical applications. Image denoising was evaluated using Gaussian and wavelet-based filters. While wavelet-based denoising improved accuracy to some extent, its increased computation time (~10 s/image) was the trade-off. The intra-retinal layer thicknesses computed by the segmentation algorithm was consistent with previous studies and demonstrated high accuracy with respect to manual segmentation, thus indicating clinical relevance. Future research will explore integrating machine learning to improve robustness across diverse retinal pathologies, enhancing the algorithm’s applicability in clinical settings. Full article

This follow-up scientific case study builds on prior research to explore the computational challenges of applying quantum algorithms to financial asset management, focusing specifically on solving the graph-cut problem for investment recommendation. Unlike our prior study, which focused on idealized QAOA performance, this work introduces a graph compression pipeline that enables QAOA deployment under real quantum hardware constraints. This study investigates quantum-accelerated spectral graph compression for financial asset recommendations, addressing scalability and regulatory constraints in portfolio management. We propose a hybrid framework combining the Quantum Approximate Optimization Algorithm (QAOA) with spectral graph theory to solve the Max-Cut problem for investor clustering. Our methodology leverages quantum simulators (cuQuantum and Cirq-GPU) to evaluate performance against classical brute-force enumeration, with graph compression techniques enabling deployment on resource-constrained quantum hardware. The results underscore that efficient graph compression is crucial for successful implementation. The framework bridges theoretical quantum advantage with practical financial use cases, though hardware limitations (qubit counts, coherence times) necessitate hybrid quantum-classical implementations. These findings advance the deployment of quantum algorithms in mission-critical financial systems, particularly for high-dimensional investor profiling under regulatory constraints. Full article

In response to the growing prevalence of tariffs as instruments of economic policy and strategic competition, this paper introduces a formal mathematical framework for optimizing counter-tariff strategies. We model the global trade ecosystem as a multi-layered, directed, weighted hypergraph, where vertices represent countries, industries, and subindustries, and hyperedges capture complex trade relationships and supply chain dependencies. The proposed framework employs bilevel optimization techniques to maximize strategic impact on target economies while minimizing self-inflicted economic costs. Through integration of graph theory, spectral analysis, and multilevel optimization methods, we develop a rigorous formalism that enables policymakers to identify optimal counter-tariff portfolios under various constraints. Our model explicitly accounts for industrial interdependencies, where export competitiveness depends on imported inputs, thus providing a more realistic representation of global value chains. Case studies applying our model to historical trade disputes demonstrate its capacity to generate superior strategic outcomes compared to conventional approaches. The framework’s axiomatic foundation allows for rapid recalibration in response to shifting economic conditions and policy objectives. Full article

This paper explores the common neighborhood energy ($E_{CN}\left(\mathsf{\Gamma }\right)$) of graphs derived from the dihedral group $D_{2n}$ and generalized quaternion group $Q_{4n}$, specifically the non-commuting graph (NCM-graph) and the commuting graph (CM-graph). Studying graphs associated with groups offers a powerful approach to translating algebraic properties into combinatorial structures, enabling the application of graph-theoretic tools to understand group behavior. The common neighborhood energy, defined as the sum of the absolute values of the eigenvalues of the common neighborhood (CN) matrix, i.e., ${\sum }_{i=1}^p\left|{\zeta }_i\right|$, where ${\left\{{\zeta }_i\right\}}_{i=1}^p$ are the CN eigenvalues, provides insights into the structural properties of these graphs. We derive explicit formulas for the CN characteristic polynomials and corresponding CN eigenvalues for both the NCM-graph and CM-graph as functions of n. Consequently, we establish closed-form expressions for the $E_{CN}$ of these graphs, which are parameterized by n. The validity of our theoretical results is confirmed through computational examples. This study contributes to the spectral analysis of algebraic graphs, demonstrating a direct connection between the group-theoretic structure of $D_{2n}$ and $Q_{4n}$, as well as the combinatorial energy of their associated graphs, thus furthering the understanding of group properties through spectral graph theory. Full article

Background: Both aging and cognitive fatigue are significant factors influencing alpha activity in the brain. However, the interactive effects of age and mental fatigue on the alpha spectrum and functional connectivity have not been fully elucidated. Methods: Using resting-state EEG data from an open-access dataset (younger: N = 198; older: N = 227) collected before and after a 2 h cognitive task block, we systematically examined the effects of aging and mental fatigue on alpha (8–13 Hz) oscillations via an aperiodic-corrected power spectrum, the weighted phase lag index (wPLI), and graph theory analysis. Results: In both spectral power and network efficiency, mental fatigue primarily modulates low alpha in younger individuals, while high alpha reflects stable age-related changes. The aperiodic offset and exponent decrease with age, while mental fatigue leads to an increase in the exponent. Notable interactions between age and mental fatigue are observed in low-alpha power, the aperiodic exponent, and the network efficiency of both low- and high-alpha bands. Conclusions: This study provides valuable insights into the differential modulation patterns of alpha activity by age and mental fatigue, as well as their interactions. These findings advance our understanding of how aging and mental fatigue differentially and interactively shape neural dynamics. Full article

This study explores the degree energy of fuzzy graphs to establish fundamental spectral bounds and characterize adjacency structures. We derive upper bounds on the sum of squared degree eigenvalues based on vertex degree distributions and formulate constraints using the characteristic polynomial of the maximum degree matrix. Furthermore, we demonstrate that the average degree energy of a connected fuzzy graph remains strictly positive. The proposed framework is applied to protein–protein interaction networks, identifying critical proteins and enhancing network resilience analysis. Full article

Even though Kemeny’s constant was first discovered in Markov chains and expressed by Kemeny in terms of mean first passage times on a graph, it can also be expressed using the pseudo-inverse of the Laplacian matrix representing the graph, which facilitates the calculation of a sharp upper bound of Kemeny’s constant. We show that for certain classes of graphs, a previously found bound is tight, which generalises previous results for bipartite and (generalised) windmill graphs. Moreover, we show numerically that for real-world networks, this bound can be used to find good numerical approximations for Kemeny’s constant. For certain graphs consisting of up to 100 K nodes, we find a speedup of a factor 30, depending on the accuracy of the approximation that can be achieved. For networks consisting of over 500 K nodes, the approximation can be used to estimate values for the Kemeny constant, where exact calculation is no longer feasible within reasonable computation time. Full article

In this study, we introduced a novel graph product derived from the standard Cartesian product and investigated its structural properties, with a particular emphasis on its independence number and spectral characteristics in relation to identical neighbor structures. A key finding is that the spectrum of this newly defined product graph consists entirely of integral eigenvalues, a significant property with applications in chemistry, network theory, and combinatorial optimization. We defined $C{N}_s$ vertices as the vertices having an identical set of neighbors and classified graphs containing such vertices as $C{N}_s$ graphs. Furthermore, we introduced the $C{N}_s$ Cartesian product for these graphs. To formally characterize the relationships between $C{N}_s$ vertices, we constructed an $n\times n$$C{N}_s$ matrix, where an entry is 1 if the corresponding pair of vertices are $C{N}_s$ vertices and 0 otherwise. Utilizing this matrix, we established that the spectrum of the $C{N}_s$ Cartesian product consists exclusively of integral eigenvalues. This finding enhances our understanding of graph spectra and their relation to structural properties. Full article

The node representation learning capability of Graph Convolutional Networks (GCNs) is fundamentally constrained by dynamic instability during feature propagation, yet existing research lacks systematic theoretical analysis of stability control mechanisms. This paper proposes a Stability-Optimized Graph Convolutional Network (SO-GCN) that enhances training stability and feature expressiveness in shallow architectures through continuous–discrete dual-domain stability constraints. By constructing continuous dynamical equations for GCNs and rigorously proving conditional stability under arbitrary parameter dimensions using nonlinear operator theory, we establish theoretical foundations. A Precision Weight Parameter Mechanism is introduced to determine critical Frobenius norm thresholds through feature contraction rates, optimized via differentiable penalty terms. Simultaneously, a Dynamic Step-size Adjustment Mechanism regulates propagation steps based on spectral properties of instantaneous Jacobian matrices and forward Euler discretization. Experimental results demonstrate SO-GCN’s superiority: 1.1–10.7% accuracy improvement on homophilic graphs (Cora/CiteSeer) and 11.22–12.09% enhancement on heterophilic graphs (Texas/Chameleon) compared to conventional GCN. Hilbert–Schmidt Independence Criterion (HSIC) analysis reveals SO-GCN’s superior inter-layer feature independence maintenance across 2–7 layers. This study establishes a novel theoretical paradigm for graph network stability analysis, with practical implications for optimizing shallow architectures in real-world applications. Full article

In this paper, a novel Doppler shift estimation method for frequency diverse array (FDA) radar based on graph signal processing (GSP) theory is proposed and investigated. First, a well-designed graph signal model for a monostatic linear FDA is formulated. Subsequently, spectral decomposition is conducted on the constructed signal model utilizing graph Fourier transform (GFT) techniques, enabling the extraction of the target’s Doppler shift parameter through spectral peak search. A comprehensive series of simulation experiments demonstrates that the proposed method can achieve the accurate estimation of target parameters even under low signal-to-noise ratio (SNR) conditions. Furthermore, the proposed method exhibits superior performance compared to the MUSIC algorithm, offering enhanced resolution and estimation accuracy. Additionally, the method is highly amenable to parallel processing, significantly reducing the computational burden associated with traditional procedures. Full article

The study of spectral graph determination is a fascinating area of research in spectral graph theory and algebraic combinatorics. This field focuses on examining the spectral characterization of various classes of graphs, developing methods to construct or distinguish cospectral nonisomorphic graphs, and analyzing the conditions under which a graph’s spectrum uniquely determines its structure. This paper presents an overview of both classical and recent advancements in these topics, along with newly obtained proofs of some existing results, which offer additional insights. Full article

We have found that the dye 1,3,3-trimethyl-2-((1′E,3′E,5′E)-5’-(1″,3″,3″-trimethylindol-(2′E)-ylidene)-penta-1″,3″-dien-1″-yl)-3H-indol-1-ium (DTMI-5) can be successfully used for the simple green determination of H₂O₂ and Fe(III)/Fe(II) species. The dye is characterized by a successful combination of spectral, protolytic, and redox properties, and the process of its decolorization in the Fenton reaction is monitored using an optical immersion probe. Theoretical calculations of the reactive sites in the DTMI-5 molecule under free radical attack reveal that the methine groups of the penta-1′,3′-dien-1′-yl linker serve as the primary reactive centers in Fe³⁺ or Fenton-type oxidation conditions. Density functional theory (DFT) calculations indicate that the redox potentials of the examined structures range from 0.34 to 1.65 eV. The experimentally observed broad peak at 340–360 nm, which appears after the interaction of DTMI-5 with the Fenton reagent, is attributed to the formation of aldehyde-type oxidation products, whose theoretical CIS(D) absorption maxima were 358.1 and 337.4 nm. The influence of various factors on the course of the reaction was experimentally investigated. The most important analytical characteristics of the methods, such as linearity intervals of calibration graphs, precision, LOD and LOQ values, selectivity coefficients, etc., were determined. The developed methods were applied to model and real samples (water, oxidation emulsion, and fertilizer). Full article

We study the algebraic connectivity for several classes of random semi-regular graphs. For large random semi-regular bipartite graphs, we explicitly compute both their algebraic connectivity as well as the full spectrum distribution. For an integer $d\in \left[3,7\right]$, we find families of random semi-regular graphs that have higher algebraic connectivity than random d-regular graphs with the same number of vertices and edges. On the other hand, we show that regular graphs beat semi-regular graphs when $d\ge 8$. More generally, we study random semi-regular graphs whose average degree is d, not necessarily an integer. This provides a natural generalization of a d-regular graph in the case of a non-integer $d$. We characterize their algebraic connectivity in terms of a root of a certain sixth-degree polynomial. Finally, we construct a small-world-type network of an average degree of 2.5 with relatively high algebraic connectivity. We also propose some related open problems and conjectures. Full article