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Different from other computer vision tasks, action recognition needs to process larger-scale video data. How to extract and analyze the effective parts from a huge amount of video information is the main difficulty of action recognition technology. In recent years, due to the outstanding performance of Graph Convolutional Networks (GCN) in many fields, a new solution to the action recognition algorithm has emerged. However, in current GCN models, the constant physical adjacency matrix makes it difficult to mine synergistic relationships between key points that are not directly connected in physical space. Additionally, a simple time connection of skeleton data from different frames makes each frame in the video contribute equally to the recognition results, which increases the difficulty of distinguishing action stages. In this paper, the information extraction ability of the model has been optimized in the space domain and time domain, respectively. In the space domain, an Adjacency Matrix Generation (AMG) module, which can pre-analyze node sets and generate an adaptive adjacency matrix, has been proposed. The adaptive adjacency matrix can help the graph convolution model to extract the synergistic information between the key points that are crucial for recognition. In the time domain, the Time Domain Attention (TDA) mechanism has been designed to calculate the time-domain weight vector through double pooling channels and complete the weights of key point sequences. Furthermore, performance of the improved TDA-AMG-GCN modules has been verified on the NTU-RGB+D dataset. Its detection accuracy at the CS and CV divisions reached 84.5% and 89.8%, respectively, with an average level higher than other commonly used detection methods at present. [ABSTRACT FROM AUTHOR]

In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too. [ABSTRACT FROM AUTHOR]

A singular graph G, defined when its adjacency matrix is singular, has important applications in mathematics, natural sciences and engineering. The chemical importance of singular graphs lies in the fact that if the molecular graph is singular, the nullity (the number of the zero eigenvalue) is greater than 0, then the corresponding chemical compound is highly reactive or unstable. By this reasoning, chemists have a great interest in this problem. Thus, the problem of characterization singular graphs was proposed and raised extensive studies on this challenging problem thereafter. The graph obtained by conglutinating the starting vertices of three paths P s 1 , P s 2 , P s 3 into a vertex, and three end vertices into a vertex on the cycle C a 1 , C a 2 , C a 3 , respectively, is denoted as γ (a 1 , a 2 , a 3 , s 1 , s 2 , s 3) . Note that δ (a 1 , a 2 , a 3 , s 1 , s 2) = γ (a 1 , a 2 , a 3 , s 1 , 1 , s 2) , ζ (a 1 , a 2 , a 3 , s) = γ (a 1 , a 2 , a 3 , 1 , 1 , s) , φ (a 1 , a 2 , a 3) = γ (a 1 , a 2 , a 3 , 1 , 1 , 1) . In this paper, we give the necessity and sufficiency that the γ − graph, δ − graph, ζ − graph and φ − graph are singular and prove that the probability that a randomly given γ − graph, δ − graph, ζ − graph or φ − graph being singular is equal to 325 512 , 165 256 , 43 64 , 21 32 , respectively. From our main results, we can conclude that such a γ − graph( δ − graph, ζ − graph, φ − graph) is singular if at least one cycle is a multiple of 4 in length, and surprisingly, the theoretical probability of these graphs being singular is more than half. This result promotes the understanding of a singular graph and may be promising to propel the solutions to relevant application problems. [ABSTRACT FROM AUTHOR]

A Modular Reconfigurable Robot (MRR) composed of standard joint and link modules has the advantages of rapid changeover of assembly configurations for a diversity of application requirements. To tackle the kinematics analysis issues for an MRR, it is essential to develop a generic method to mathematically represent all possible assembly configurations and automatically generate their kinematic models. In this paper, two types of robot modules, i.e., revolute joint modules and rigid link modules, are considered as the basic building blocks of an MRR. The topological structure of an MRR is represented by a graph, termed an Assembly Graph (AG), in which all robot modules, regardless of their types, are treated as vertices, while the connecting interfaces between any two robot modules are treated as edges. Based on graph theory, a modified adjacency matrix, termed an Assembly Adjacency Matrix (AAM), is proposed to represent the assembly configuration of an MRR, in which the three-dimensional assembly information is specified with connecting port vectors for the adjacent modules. A path matrix derived from the AAM of an MRR is employed to describe the connecting sequence of its constituting modules. The local frame representation of the Product-of-Exponentials (POE) formula is employed, which is configuration-independent. Therefore, for an MRR configuration represented with an AAM, its kinematic model will be automatically generated according to its path matrix. The proposed assembly configuration representation and kinematic modeling method are validated through a dual-branch MRR configuration. [ABSTRACT FROM AUTHOR]

The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal multi-ring mixed (CMRM) graphs. In the case of mixed graphs, we can have edges (without direction) and arcs (with direction). The chordal ring and chordal ring mixed graphs are bipartite and 3-regular. They consist of a number r (for r ≥ 1 ) of (undirected or directed) cycles with some edges (the chords) joining them. In particular, for CMR, when r = 1 , that is, with only one undirected cycle, we obtain the known families of chordal ring graphs. Here, we used plane tessellations to represent our chordal multi-ring graphs. This allowed us to obtain their maximum number of vertices for every given diameter. Additionally, we computationally obtained their minimum diameter for any value of the number of vertices. Moreover, when seen as a lift graph (also called voltage graph) of a base graph on Abelian groups, we obtained closed formulas for the spectrum, that is, the eigenvalue multi-set of its adjacency matrix. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple connected graph with vertex set V and edge set E, respectively. The term "anti-reciprocal eigenvalue property" refers to a non-singular graph G for which, − 1 λ ∈ σ (G) , whenever λ ∈ σ (G) , ∀ λ ∈ σ (G) . Here, σ (G) is the multiset of all eigenvalues of A (G) . Moreover, if multiplicities of eigenvalues and their negative reciprocals are equal, then that graph is said to have strong anti-reciprocal eigenvalue properties, and the graph is referred to as a strong anti-reciprocal graph (or (− S R) graph). In this article, a new family of graphs F n (k , j) is introduced and the energy of F 5 (k , k 2) k ≥ 2 is calculated. Furthermore, with the help of F 5 (k , k 2) , some families of (− S R) graphs are constructed. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the double domination in vague graphs, and it is a contribution to the Special Issue "Advances in graph theory and Symmetry/Asymmetry" of Symmetry. Symmetry is one of the most important criteria that illustrate the structure and properties of fuzzy graphs. It has many applications in dominating sets and helps find a suitable place for construction. Vague graphs (VGs), which are a family of fuzzy graphs (FGs), are a well-organized and useful tool for capturing and resolving a range of real-world scenarios involving ambiguous data. In the graph theory, a dominating set (DS) for a graph G * = (X , E) is a subset D of the vertices X so that every vertex which is not in D is adjacent to at least one member of D. The subject of energy in graph theory is one of the most attractive topics serving a very important role in biological and chemical sciences. Hence, in this work, we express the notion of energy on a dominating vague graph (DVG) and also use the concept of energy in modeling problems related to DVGs. Moreover, we introduce a new notion of a double dominating vague graph (DDVG) and provide some examples to explain various concepts introduced. Finally, we present an application of energy on DVGs. [ABSTRACT FROM AUTHOR]

The A γ matrix of a graph G is determined by A γ (G) = (1 − γ) A (G) + γ D (G) , where 0 ≤ γ ≤ 1 , A (G) and D (G) are the adjacency and the diagonal matrices of node degrees, respectively. In this case, the A γ matrix brings together the spectral theories of the adjacency, the Laplacian, and the signless Laplacian matrices, and many more γ adjacency-type matrices. In this paper, we obtain the A γ eigenvalues of zero divisor graphs of the integer modulo rings and the von Neumann rings. These results generalize the earlier published spectral theories of the adjacency, the Laplacian and the signless Laplacian matrices of zero divisor graphs. [ABSTRACT FROM AUTHOR]

Accurate and real-time traffic speed prediction remains challenging due to the irregularity and asymmetry of real-traffic road networks. Existing models based on graph convolutional networks commonly use multi-layer graph convolution to extract an undirected static adjacency matrix to map the correlation of nodes, which ignores the dynamic symmetry change of correlation over time and faces the challenge of oversmoothing during training iterations, making it difficult to learn the spatial structure and temporal trend of the traffic network. To overcome the above challenges, we propose a novel multi-head self-attention gated spatiotemporal graph convolutional network (MSGSGCN) for traffic speed prediction. The MSGSGCN model mainly consists of the Node Correlation Estimator (NCE) module, the Time Residual Learner (TRL) module, and the Gated Graph Convolutional Fusion (GGCF) module. Specifically, the NCE module aims to capture the dynamic spatiotemporal correlations between nodes. The TRL module utilizes a residual structure to learn the long-term temporal features of traffic data. The GGCF module relies on adaptive diffusion graph convolution and gated recurrent units to learn the key spatial features of traffic data. Experimental analysis on a pair of real-world datasets indicates that the proposed MSGSGCN model enhances prediction accuracy by more than 4% when contrasted with state-of-the-art models. [ABSTRACT FROM AUTHOR]

Coronavirus Disease 2019 (COVID-19) continues to spread throughout the world, and it is necessary for us to implement effective methods to prevent and control the spread of the epidemic. In this paper, we propose a new model called Spatial–Temporal Attention Graph Convolutional Networks (STAGCN) that can analyze the long-term trend of the COVID-19 epidemic with high accuracy. The STAGCN employs a spatial graph attention network layer and a temporal gated attention convolutional network layer to capture the spatial and temporal features of infectious disease data, respectively. While the new model inherits the symmetric "space-time space" structure of Spatial–Temporal Graph Convolutional Networks (STGCN), it enhances its ability to identify infectious diseases using spatial–temporal correlation features by replacing the graph convolutional network layer with a graph attention network layer that can pay more attention to important features based on adaptively adjusted feature weights at different time points. The experimental results show that our model has the lowest error rate compared to other models. The paper also analyzes the prediction results of the model using interpretable analysis methods to provide a more reliable guide for the decision-making process during epidemic prevention and control. [ABSTRACT FROM AUTHOR]

For a vertex v in a graph G, the open support of v under addition is the sum of degrees of all its neighbors. The open support of G under addition is the sum of open supports of all its vertices. The results for open support of graphs are deeply dependent on the structure property of the graph considered, such as its symmetry. In this paper, we generalize the concept of open support to hypergraphs. We give some examples and prove a general formula for open support of hypergraphs by using the adjacency and incidence matrices method. [ABSTRACT FROM AUTHOR]

Let D be a digraph with n vertices and a arcs. The Laplacian and the signless Laplacian matrices of D are, respectively, defined as L (D) = D e g + (D) − A (D) and Q (D) = D e g + (D) + A (D) , where A (D) represents the adjacency matrix and D e g + (D) represents the diagonal matrix whose diagonal elements are the out-degrees of the vertices in D. We derive a combinatorial representation regarding the first few coefficients of the (signless) Laplacian characteristic polynomial of D. We provide concrete directed motifs to highlight some applications and implications of our results. The paper is concluded with digraph examples demonstrating detailed calculations. [ABSTRACT FROM AUTHOR]

In this note, I derive simple formulas based on the adjacency matrix of a network to compute measures associated with Ronald S. Burt's structural holes (effective size, redundancy, local constraint, and constraint), together with the measure called improved structural holes introduced in 2017. This can help to see these measures within a unified computation framework because they can all be expressed in matricial form. These formulas can also be used to define naïve algorithms based on matrix operations for their computation. Such naïve algorithms can be used for small- and medium-sized networks, where exploiting the sparsity of the matrices and efficient triangle listing techniques are not necessary. [ABSTRACT FROM AUTHOR]

Link prediction is one of the most important and challenging tasks in complex network analysis, which aims to predict the existence of unknown links based on the known information in the network. As critical topological properties in the network, node's degree and clustering coefficient are well-suited for describing the tightness of connection between nodes. The importance of node can affect the possibility of link existence to a certain extent. By analyzing the impact of different centrality on links, which concluded that the degree centrality and proximity centrality have the greatest influence on network link prediction. A link prediction algorithm combines importance of node and network topological properties, called DCCLP, is proposed in this paper, the symmetry of the adjacency matrix is considered in the DCCLP link prediction algorithm to further describe the structural similarity of network nodes. In the training phase of the DCCLP algorithm, the maximized AUC indicator in the training set as the objective, and the optimal parameters are estimated by utilizing the White Shark Optimization algorithm. Then the prediction accuracy of the DCCLP algorithm is evaluated in the test set. By experimenting on twenty-one networks with different scales, and comparing with existing algorithms, the experimental results show that the effectiveness and feasibility of DCCLP algorithm, and further illustrate the importance of the degree centrality of node pairs and proximity centrality of nodes to improve the prediction accuracy of link prediction. [ABSTRACT FROM AUTHOR]

A graph containing some edges with probability measures and other edges with uncertain measures is referred to as an uncertain random graph. Numerous real-world problems in social networks and transportation networks can be boiled down to optimization problems in uncertain random graphs. Actually, information in optimization problems in uncertain random graphs is always asymmetric. Regularization is a common optimization problem in graph theory, and the regularity index is a fundamentally measurable indicator of graphs. Therefore, this paper investigates the regularity index of an uncertain random graph within the framework of chance theory and information asymmetry theory. The concepts of k-regularity index and regularity index of the uncertain random graph are first presented on the basis of the chance theory. Then, in order to compute the k-regularity index and the regularity index of the uncertain random graph, a simple and straightforward calculating approach is presented and discussed. Furthermore, we discuss the relationship between the regularity index and the k-regularity index of the uncertain random graph. Additionally, an adjacency matrix-based algorithm that can compute the k-regularity index of the uncertain random graph is provided. Some specific examples are given to illustrate the proposed method and algorithm. Finally, we conclude by highlighting some potential applications of uncertain random graphs in social networks and transportation networks, as well as the future vision of its combination with symmetry. [ABSTRACT FROM AUTHOR]

Vague graphs (VGs), belonging to the fuzzy graphs (FGs) family, have good capabilities when faced with problems that cannot be expressed by FGs. The notion of a VG is a new mathematical attitude to model the ambiguity and uncertainty in decision-making issues. A vague fuzzy graph structure (VFGS) is the generalization of the VG. It is a powerful and useful tool to find the influential person in various relations. VFGSs can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problems where fuzzy graphs may fail to reveal satisfactory results. Moreover, VGSs are very useful tools for the study of different domains of computer science such as networking, social systems, and other issues such as bioscience and medical science. The subject of energy in graph theory is one of the most attractive topics that is very important in biological and chemical sciences. Hence, in this work, we extend the notion of energy of a VG to the energy of a VFGS and also use the concept of energy in modeling problems related to VFGS. Actually, our purpose is to develop a notion of VFGS and investigate energy and Laplacian energy (LE) on this graph. We define the adjacency matrix (AM) concept, energy, and LE of a VFGS. Finally, we present three applications of the energy in decision-making problems. [ABSTRACT FROM AUTHOR]

A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of parameters to determine a graph uniquely. Other related inverse problems, including the Polynomial Reconstruction Problem (PRP), involve the recovery of graph invariants. The PRP seeks to extract the spectrum of a graph from the deck of cards each showing the spectrum of a vertex-deleted subgraph. We show how various algebraic methods join forces to reconstruct a graph or its invariants from a minimal set of restricted eigenvalue-eigenvector information of the parent graph or its subgraphs. We show how functions of the entries of eigenvectors of the adjacency matrix A of a graph can be retrieved from the spectrum of eigenvalues of A. We establish that there are two subclasses of disconnected graphs with each card of the deck showing a common eigenvalue. These could occur as possible counter examples to the positive solution of the PRP. [ABSTRACT FROM AUTHOR]

Let G be a graph with n vertices, let A (G) be an adjacency matrix of G and let P A (G , λ) be the characteristic polynomial of A (G) . The adjacency spectrum of G consists of eigenvalues of A (G) . A graph G is said to be determined by its adjacency spectrum (DS for short) if other graphs with the same adjacency spectrum as G are isomorphic to G. In this paper, we investigate the spectral characterization of unicycle graphs with only two vertices of degree three. We use G 21 (s 1 , s 2) to denote the graph obtained from Q (s 1 , s 2) by identifying its pendant vertex and the vertex of degree two of P 3 , where Q (s 1 , s 2) is the graph obtained by identifying a vertex of C s 1 and a pendant vertex of P s 2 . We use G 31 (t 1 , t 2) to denote the graph obtained from circle with the vertices v 0 v 1 ⋯ v t 1 + t 2 + 1 by adding one pendant edge at vertices v 0 and v t 1 + 1 , respectively. It is shown that G 21 (s 1 , s 2) ( s 1 ≠ 4 , 6 , s 1 ≥ 3 , s 2 ≥ 3 ) and G 31 (t 1 , t 2) ( t 1 + t 2 ≠ 2 , t 2 ≥ t 1 ≥ 1 ) are determined by their adjacency spectrum. [ABSTRACT FROM AUTHOR]

The designer network is the carrier for designers to carry out product design tasks, and the study of its evolution helps to accurately identify the designer team. The uncertainty of whether the designer chooses to keep in touch with other members of the network, its nonlinearity on the overall performance of the network and the external interference of the network, etc., increase the complexity of the designer network, which is not conducive to the development of product design tasks. Therefore, it is very interesting and important to study the co-evolution mechanism of the designer's network structure and the designer's opinion. This paper analyzes the Deffuant opinion dynamics model, considers the asymmetry of nodes in the network caused by the designer's unequal relationship (the number of node links is different), and the difference in the degree of acceptance of other people's opinions by different individuals, and improves the Deffuant model. Then combining the improved Deffuant model with the BA (Barabasi–Albert) model, a DBA (Deffuant and BA Adaptive) model was proposed that integrates opinion update, broken edge and reconnection, and opinion changes. On this basis, this paper designs a virtual network through the crowdsourcing of a certain product. The adjacency matrix of this network is symmetric, and corresponding comparative experiments are carried out on this network. The analysis of test results shows that under six different deliberate attacks, the DBA model is more robust than the BA model. In addition, the average shortest path of the DBA network will vary with the parameters. The proposed integrated DBA model has important guiding significance for building a robust designer network. [ABSTRACT FROM AUTHOR]

The aim of the present work is a comparative study of different persistence kernels applied to various classification problems. After some necessary preliminaries on homology and persistence diagrams, we introduce five different kernels that are then used to compare their performances of classification on various datasets. We also provide the Python codes for the reproducibility of results and, thanks to the symmetry of kernels, we can reduce the computational costs of the Gram matrices. [ABSTRACT FROM AUTHOR]

The study delves into the relationship between symmetry groups and automorphism groups in polyhedral graphs, emphasizing their interconnected nature and their significance in understanding the symmetries and structural properties of fullerenes. It highlights the visual importance of symmetry and its applications in architecture, as well as the mathematical structure of the automorphism group, which captures all of the symmetries of a graph. The paper also discusses the significance of groups in Abstract Algebra and their relevance to understanding the behavior of mathematical systems. Overall, the findings offer an inclusive understanding of the relationship between symmetry groups and automorphism groups, paving the way for further research in this area. [ABSTRACT FROM AUTHOR]

To address the symmetry-related resilience issues of stations and lines in urban rail transit networks, we propose a two-stage robust optimization-based approach for urban rail transit network planning. In this context, resilience is conceptualized as the ability of the network to maintain its operational symmetry under normal and disruptive conditions. Firstly, we used passenger flow distributions as decision variables to construct a two-stage symmetry-based urban rail transit network planning model, aiming to simultaneously minimize the total cost and total operating time of the network while preserving its functional symmetry. Secondly, we designed a hybrid evolutionary algorithm with chromosomes having a two-layer encoding structure, where the Niched Pareto Genetic Algorithm served as the main algorithmic framework, and a Large Neighborhood Search mechanism was designed to optimize the connectivity gene layer of individuals, ensuring the symmetry of network connectivity. Finally, we conducted computational verification on randomly generated instances to confirm the effectiveness of the model and algorithm. The experimental results demonstrated that our method could find two sets of Pareto optimal solutions for cost preference and time preference, thereby preserving the operational symmetry of the network under normal and damaged conditions, as well as reducing the total operating time. This effectively improved the overall efficiency and resilience of the network. Our designed hybrid evolutionary algorithm converged to satisfactory objective values in the early iterations, exhibiting strong search and optimization performance and effectively solving the two-stage symmetry-based urban rail transit network planning model. [ABSTRACT FROM AUTHOR]

This paper proposes the complex dynamics of collective behavior through an interdisciplinary approach that integrates individual cognition with potential fields. Firstly, the interaction between individual cognition and external potential fields in complex social systems is explored, integrating perspectives from physics, cognitive psychology, and social science. Subsequently, a new modeling method for the multidimensional potential field mechanism is proposed, aiming to reduce individual behavioral errors and cognitive dissonance, thereby improving system efficiency and accuracy. The approach uses cooperative control and distributed learning algorithms to simulate collective behavior, allowing individuals to iteratively adapt based on local information and collective intelligence. Simulations highlight the impact of factors such as individual density, noise intensity, communication radius, and negative potential fields on collective dynamics. For instance, in a high-density environment with 180 individuals, increased social friction and competition for resources significantly decrease collective search efficiency. Validation is achieved by comparing simulation results with existing research, showing consistency and improvements over traditional models. In noisy environments, simulations maintain higher accuracy and group cohesion compared to standard methods. Additionally, without communication, the Mean Squared Error (MSE) initially drops rapidly as individuals adapt but stabilizes over time, emphasizing the importance of communication in maintaining collective efficiency. The study concludes that collective behavior emerges from complex nonlinear interactions between individual cognition and potential fields, rather than being merely the sum of individual actions. These insights enhance the understanding of complex system dynamics, providing a foundation for future applications in adaptive urban environments and the design of autonomous robots and AI systems. [ABSTRACT FROM AUTHOR]

Despite recent progress in the study of complex systems, reconstruction of damaged networks due to random and targeted attack has not been addressed before. In this paper, we formulate the network reconstruction problem as an identification of network structure based on much reduced link information. Furthermore, a novel method based on multilayer perceptron neural network is proposed as a solution to the problem of network reconstruction. Based on simulation results, it was demonstrated that the proposed scheme achieves very high reconstruction accuracy in small-world network model and a robust performance in scale-free network model. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to address the challenge of designing finite-time fault-tolerant control mechanisms for nonlinear flexible spacecraft systems, which are particularly vulnerable to randomized actuator faults. Diverging from traditional methodologies, our research harnesses the capabilities of the Takagi–Sugeno (T–S) fuzzy framework. A unique feature of our model is the representation of actuator failures as stochastic signals following a Markov process, thereby offering a robust solution for addressing timeliness concerns. In this paper, we introduce a generalized reciprocally convex inequality that includes adjustable parameters, broadening the scope of previous results by accommodating them as special cases. Through the amalgamation of this enhanced inequality and flexible independent parameters, we propose an innovative controller design strategy. This approach establishes a stability standard that guarantees mean-square H ∞ performance. In order to validate the efficacy of the suggested strategy, we present a numerical illustration involving a nonlinear spacecraft system, showcasing the practical advantages and feasibility of our proposed technique. [ABSTRACT FROM AUTHOR]

Mainstream semi-supervised learning (SSL) techniques, such as pseudo-labeling and contrastive learning, exhibit strong generalization abilities but lack theoretical understanding. Furthermore, pseudo-labeling lacks the label enhancement from high-quality neighbors, while contrastive learning ignores the supervisory guidance provided by genuine labels. To this end, we first introduce a generalized bias-variance decomposition framework to investigate them. Then, this research inspires us to propose two new techniques to refine them: neighbor-enhanced pseudo-labeling, which enhances confidence-based pseudo-labels by incorporating aggregated predictions from high-quality neighbors; label-enhanced contrastive learning, which enhances feature representation by combining enhanced pseudo-labels and ground-truth labels to construct a reliable and complete symmetric adjacency graph. Finally, we combine these two new techniques to develop an excellent SSL method called GBVSSL. GBVSSL significantly surpasses previous state-of-the-art SSL approaches in standard benchmarks, such as CIFAR-10/100, SVHN, and STL-10. On CIFAR-100 with 400, 2500, and 10,000 labeled samples, GBVSSL outperforms FlexMatch by 3.46%, 2.72%, and 2.89%, respectively. On the real-world dataset Semi-iNat 2021, GBVSSL improves the Top-1 accuracy over CCSSL by 4.38%. Moreover, GBVSSL exhibits faster convergence and enhances unbalanced SSL. Extensive ablation and qualitative studies demonstrate the effectiveness and impact of each component of GBVSSL. [ABSTRACT FROM AUTHOR]

Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages. [ABSTRACT FROM AUTHOR]

This study aims to address the issue of redundancy and interference in data-collection systems by proposing a novel feature-selection method based on maximum information coefficient (MIC) and graph symmetry structure in complex-network theory. The method involves establishing a weighted feature network, identifying key features using dominance set and node strength, and employing the binary particle-swarm algorithm and LS-SVM algorithm for solving and validation. The model is implemented on the UNSW-NB15 and UCI datasets, demonstrating noteworthy results. In comparison to the prediction methods within the datasets, the model's running speed is significantly reduced, decreasing from 29.8 s to 6.3 s. Furthermore, when benchmarked against state-of-the-art feature-selection algorithms, the model achieves an impressive average accuracy of 90.3%, with an average time consumption of 6.3 s. These outcomes highlight the model's superiority in terms of both efficiency and accuracy. [ABSTRACT FROM AUTHOR]

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we study the way the symmetries of a given graph are reflected in its characteristic polynomials. Our aim is not only to find obstructions for graph symmetries in terms of its polynomials but also to measure how faithful these algebraic invariants are with respect to symmetry. Let p be an odd prime and Γ be a finite graph whose automorphism group contains an element h of order p. Assume that the finite cyclic group generated by h acts semi-freely on the set of vertices of Γ with fixed set F. We prove that the characteristic polynomial of Γ , with coefficients in the finite field of p elements, is the product of the characteristic polynomial of the induced subgraph Γ [ F ] by one of Γ \ F . A similar congruence holds for the characteristic polynomial of the Laplacian matrix of Γ. [ABSTRACT FROM AUTHOR]

The reliability of circuit systems is primarily affected by cascading failures due to their complex structural and functional coupling. Causes of cascading failure during circuit operation include the continuous degradation process of components and external random shocks. Circuit systems can exhibit asymmetric structural changes and functional loss during cascading failure propagation due to the coupling of degradation and shock and their uncertainty effects. To tackle this issue, this paper abstracts the circuit into an impedance network and constructs a component failure behavior model that considers the correlation between degradation and shock. The interactions between soft and hard failure processes among different components are discussed. Two types of cascading failure propagation processes are described: slow propagation associated with continuous degradation and damage shock, and fast propagation due to fatal shock. Based on this, a cascading failure simulation algorithm is developed. This article presents a case study to demonstrate the proposed models and to analyze the reliability of a typical circuit system. [ABSTRACT FROM AUTHOR]

Graph neural networks (GNNs) have shown significant success in graph representation learning. However, the performance of existing GNNs degrades seriously when their layers deepen due to the over-smoothing issue. The node embedding incline converges to a certain value when GNNs repeat, aggregating the representations of the receptive field. The main reason for over-smoothing is that the receptive field of each node tends to be similar as the layers increase, which leads to different nodes aggregating similar information. To solve this problem, we propose an adaptive multi-channel deep graph neural network (AMD-GNN) to adaptively and symmetrically aggregate information from the deep receptive field. The proposed model ensures that the receptive field of each node in the deep layer is different so that the node representations are distinguishable. The experimental results demonstrate that AMD-GNN achieves state-of-the-art performance on node classification tasks with deep models. [ABSTRACT FROM AUTHOR]

We have developed novel polynomials called delta polynomials, which are, in turn, derived from the characteristic and matching polynomials of graphs associated with polycyclic aromatic compounds. Natural logarithmic aromatic indices are derived from these delta polynomials, which are shown to provide new insights into the aromaticity of polycyclic aromatic compounds, including the highly symmetric C60 buckminsterfullerene, several other fullerenes, graphene, kekulene series and other cycloarenes, such as polycyclic circumcoronaphenes and coronoids. The newly developed aromatic index yields a value of 6.77 for graphene, 6.516865 for buckminsterfullerene C60(Ih), 5.914023 for kekulene (D6h symmetry), 6.064420 for coronene (D6h), 6.137828 for circumcoronene (D6h), 6.069668 for dicronylene and so forth. Hence, the novel scaled logarithmic aromatic delta indices developed here appear to provide good quantitative measures of aromaticity, especially when they are used in conjunction with other aromatic indicators. [ABSTRACT FROM AUTHOR]

Money laundering is an illicit activity that seeks to conceal the nature and origins of criminal proceeds, posing a substantial threat to the national economy, the political order, and social stability. To scientifically and reasonably predict money laundering risks, this paper focuses on the "layering" stage of the money laundering process in the field of supervised learning for money laundering fraud prediction. A money laundering and fraud prediction model based on deep learning, referred to as MDGC-LSTM, is proposed. The model combines the use of a dynamic graph convolutional network (MDGC) and a long short-term memory (LSTM) network to efficiently identify illegal money laundering activities within financial transactions. MDGC-LSTM constructs dynamic graph snapshots with symmetrical spatiotemporal structures based on transaction information, representing transaction nodes and currency flows as graph nodes and edges, respectively, and effectively captures the relationships between temporal and spatial structures, thus achieving the dynamic prediction of fraudulent transactions. The experimental results demonstrate that compared with traditional algorithms and other deep learning models, MDGC-LSTM achieves significant advantages in comprehensive spatiotemporal feature modeling. Specifically, based on the Elliptic dataset, MDGC-LSTM improves the Macro-F1 score by 0.25 compared to that of the anti-money laundering fraud prediction model currently considered optimal. [ABSTRACT FROM AUTHOR]

Nowadays, deep representations have gained significant attention due to their outstanding performance in a wide range of tasks. However, the interpretability of deep representations in specific applications poses a significant challenge. For instances where the generated quantity matrices exhibit symmetry, this paper introduces a variant of deep matrix factorization (deep MF) called deep L p smooth symmetric non-negative matrix factorization (DSSNMF), which aims to improve the extraction of clustering structures inherent in complex hierarchical and graphical representations in high-dimensional datasets by improving the sparsity of the factor matrices. We successfully applied DSSNMF to synthetic datasets as well as datasets related to post-traumatic stress disorder (PTSD) to extract several hierarchical communities. Specifically, we identified non-disjoint communities within the partial correlation networks of PTSD psychiatric symptoms, resulting in highly meaningful clinical interpretations. Numerical experiments demonstrate the promising applications of DSSNMF in fields like network analysis and medicine. [ABSTRACT FROM AUTHOR]

In this article, a robust index named first-order network coherence (FONC) for the multi-agent systems (MASs) with layered lattice-like structure is studied via the angle of the graph spectra theory. The union operation of graphs is utilized to construct two pairs of non-isomorphic layered lattice-like structures, and the expression of the index is acquired by the approach of Laplacian spectra, then the corresponding asymptotic results are obtained. It is found that when the cardinality of the node sets of coronary substructures with better connectedness tends to infinity, the FONC of the whole network will have the same asymptotic behavior with the central lattice-like structure in the considered classic graph frameworks. The indices of the networks were simulated to illustrate the the asymptotic results, as described in the last section. [ABSTRACT FROM AUTHOR]

Identifying influential nodes, with pivotal roles in practical domains like epidemic management, social information dissemination optimization, and transportation network security enhancement, is a critical research focus in complex network analysis. Researchers have long strived for rapid and precise identification approaches for these influential nodes that are significantly shaping network structures and functions. The recently developed SPON (sum of proportion of neighbors) method integrates information from the three-hop neighborhood of each node, proving more efficient and accurate in identifying influential nodes than traditional methods. However, SPON overlooks the heterogeneity of neighbor information, derived from the asymmetry properties of natural networks, leading to its lower accuracy in identifying essential nodes. To sustain the efficiency of the SPON method pertaining to the local method, as opposed to global approaches, we propose an improved local approach, called the SSPN (sum of the structural proportion of neighbors), adapted from the SPON method. The SSPN method classifies neighbors based on the h-index values of nodes, emphasizing the diversity of asymmetric neighbor structure information by considering the local clustering coefficient and addressing the accuracy limitations of the SPON method. To test the performance of the SSPN, we conducted simulation experiments on six real networks using the Susceptible–Infected–Removed (SIR) model. Our method demonstrates superior monotonicity, ranking accuracy, and robustness compared to seven benchmarks. These findings are valuable for developing effective methods to discover and safeguard influential nodes within complex networked systems. [ABSTRACT FROM AUTHOR]

The prediction of mechanical properties of cold-rolled steel is very important for the quality control, process optimization, and cost control of cold-rolled steel, but it is still a challenging task to predict accurately. For the existing graph structure of graph attention networks, it is difficult to effectively establish the complex coupling relationship and nonlinear causal relationship between variables. At the same time, it is considered that the process of cold-rolled steel has typical full-flow process characteristics and the graph attention network makes it difficult to extract the path information between the central node and its higher-order neighborhood. The neural Granger causality algorithm is used to extract the latent relationship between variables, and the basic graph structure of mechanical property prediction data is constructed. Secondly, the node embedding layer is added before the graph attention network, which leverages the symmetry nature of Node2vec method by incorporating both breadth-first and depth-first exploration strategies. This ensures a balanced exploration of diverse paths in the graph, capturing not only local structures but also higher-order relationships. The combined graph attention networks are then able to effectively capture the symmetry path information between nodes and dependencies between variables. The accuracy and superiority of this method are verified by experiments in real cold-rolled steel production cases. [ABSTRACT FROM AUTHOR]

In the paper, we study the robust synchronization of complex dynamic networks (CDNs) with deviating arguments and parameter uncertainties via self-feedback control, the model involves both advanced and delayed arguments. In addition, based on the Gronwall inequality and inequality techniques, we derive upper bounds on the length of the arguments and the magnitude of the parameters, when the parameters and arguments of CDNs are below the upper bounds, the CDNs continue to exhibit exponential synchronization. In comparison to linear matrix inequalities and Lyapunov's method in the existing literature, we obtain elaborate bounds. Finally, Several simple examples can demonstrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

Since it is difficult for lateral stiffness of rope-guided rails to meet industry criteria in deep construction shaft, schematic diagrams of displacement suppression mechanisms (DSMs) are designed with a systematic approach demonstrated to reduce the lateral displacement of rope-guided rails in this paper. DSMs are simplified as planar four-bar and six-bar topological graphs based on topological theory. Each corresponding mechanical chain of these four-bar and six-bar mechanisms is divided into a rack, mechanical parts, prismatic, and revolute joints. An extended adjacency matrix is defined to represent the rack position, specific types of kinematic joints, and adjacency relationships between kinematic parts. Then, a symmetric vertex identification method is proposed with regard to planar 1-DOF (one degree of freedom) four-bar and six-bar topological graphs to get the sequences of prismatic joints for kinematic chains of DSMs. Finally, the alternative schematic diagrams of DSMs are obtained. The results show four-bar mechanisms with simple structure; few kinematical parts but less resident force are suitable for a mine shaft with small space and small swing. Six-bar mechanisms with two prismatic joints and three mechanical rack degree are applicable for wide shaft space in deep shaft, due to their stable structure and double resistant force. This development is helpful for DSM dimension synthesis design in future. [ABSTRACT FROM AUTHOR]

The core–periphery structure is one of the key concepts in the structural analysis of complex networks. It consists of a partitioning of the node set of a given graph or network into two groups, called core and periphery, where the core nodes induce a well-connected subgraph and share connections with peripheral nodes, while the peripheral nodes are loosely connected to the core nodes and other peripheral nodes. We propose a polynomial-time algorithm to detect core–periphery structures in networks having a symmetric adjacency matrix. The core set is defined as the solution of a combinatorial optimization problem, which has a pleasant symmetry with respect to graph complementation. We provide a complete description of the optimal solutions to that problem and an exact and efficient algorithm to compute them. The proposed approach is extended to networks with loops and oriented edges. Numerical simulations are carried out on both synthetic and real-world networks to demonstrate the effectiveness and practicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Let n ≥ 1 be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of Z n , where n = p α , p 2 q , p 2 q 2 , p q r , p 3 q , p 2 q r , and p q r s for the different prime integers p , q , r , and s. Moreover, we construct M-polynomials and C o M -polynomials using the P I S -graph structure of Z n to avoid the difficulty of computing the descriptors via formulas directly. Furthermore, we present a geometric comparison for representations of each surface obtained by M-polynomials and C o M -polynomials. Finally, we discuss the applicability of algebraic graphs to chemical graph theory. [ABSTRACT FROM AUTHOR]

Many problems of practical interest can be modeled and solved by using interval-valued fuzzy graph (IVFG) algorithms. An IVFG is a very useful and effective tool for studying various calculations, fields of intelligence, and computer science, such as networking, imaging, and other fields, such as biological sciences. In different applications, they present an appropriate construction means. There were limitations in the definition of fuzzy graphs (FGs), which prompted us to propose a new definition for IVFGs. Some interesting properties related to the new IVFGs are investigated, and enough conditions under which the level graph on IVFGs is equivalent are obtained. Therefore, in this study, we present the properties of a level graph (LG) of an IVFG, and four operations, the Cartesian product (CP), composition (CO), union, and join, are investigated on it. Today, in a treatment system, one of the issues that can be very valuable and important to the quality of service to patients is finding qualified and efficient people in each department, which is not an easy task. But the interval-valued fuzzy graph, as an important fuzzy graph, can help us by considering the ability of each person in the form of intervals of numbers and the effectiveness of each one on the other (according to the relationships between them) in order to find the most worthy people. So, an application of IVFG to find the most effective person in a hospital information system has been introduced. [ABSTRACT FROM AUTHOR]

Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel's, Helmholtz's, Laplace's, Legendre's, Poisson's, and Schrödinger's equations. The algorithm extracting the first principal component is also provided. [ABSTRACT FROM AUTHOR]

This paper proposes a new cover selection method for steganography. We focus on the scenario that the available images for selection contain diversified sources, i.e., nature images and metaverse images. For the scenario, we design a targeted strategy to evaluate the suitability for steganography of a candidate image, which selects images according to the undetectability against steganalytic tools symmetrically. Firstly, steganalytic features of the candidate images are extracted. Then, the features are fed on a steganalytic classifier, and the possibility of carrying secret data is calculated for cover selection. As a result, the selected images are the best candidates to resist steganalysis. Experimental results show that our method performs better than existing cover selection schemes when checked by steganalytic tools. [ABSTRACT FROM AUTHOR]

In this paper, in the environment of Euclidean Jordan algebras, we establish some inequalities over the Krein parameters of a symmetric association scheme and of a strongly regular graph. Next, we define the modified Krein parameters of a strongly regular graph and establish some admissibility conditions over these parameters. Finally, we introduce some relations over the Krein parameters of a strongly regular graph. [ABSTRACT FROM AUTHOR]

The maximum clique problem is a problem that takes many forms in optimization and related graph theory problems, and also has many applications. Because of its NP-completeness (nondeterministic polynomial time), the question arises of its solvability for larger instances. Instead of the traditional approaches based on the use of approximate or stochastic heuristic methods, we focus here on the use of integer programming models in the GAMS (General Algebraic Modelling System) environment, which is based on exact methods and sophisticated deterministic heuristics incorporated in it. We propose modifications of integer models, derive their time complexities and show their direct use in GAMS. GAMS makes it possible to find optimal solutions to the maximum clique problem for instances with hundreds of vertices and thousands of edges within minutes at most. For extremely large instances, good approximations of the optimum are given in a reasonable amount of time. A great advantage of this approach over all the mentioned algorithms is that even if GAMS does not find the best known solution within the chosen time limit, it displays its value at the end of the calculation as a reachable bound. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to derive a new perturbation theory (PT) that has converging series. Such series arise in the nonlocal scalar quantum field theory (QFT) with fractional power potential. We construct a PT for the generating functional (GF) of complete Green functions (including disconnected parts of functions) Z j as well as for the GF of connected Green functions G j = ln Z j in powers of coupling constant g. It has infrared (IR)-finite terms. We prove that the obtained series, which has the form of a grand canonical partition function (GCPF), is dominated by a convergent series—in other words, has majorant, which allows for expansion beyond the weak coupling g limit. Vacuum energy density in second order in g is calculated and researched for different types of Gaussian part S 0 [ ϕ ] of the action S [ ϕ ] . Further in the paper, using the polynomial expansion, the general calculable series for G j is derived. We provide, compare and research simplifications in cases of second-degree polynomial and hard-sphere gas (HSG) approximations. The developed formalism allows us to research the physical properties of the considered system across the entire range of coupling constant g, in particular, the vacuum energy density. [ABSTRACT FROM AUTHOR]

Counting polynomials find their way into chemical graph theory through quantum chemistry in two ways: as approximate solutions to the Schrödinger equation or by storing information in a mathematical form and trying to find a pattern in the roots of these expressions. Coefficients count how many times a property occurs, and exponents express the extent of the property. They help understand the origin of regularities in the chemistry of specific classes of compounds. Our objective is to accelerate the research of newcomers into chemical graph theory. One problem in understanding these concepts is in the different approaches and notations of each research study; some researchers provide online tools for computing these mathematical concepts, but these need to be maintained for functionality. We take advantage of similar mathematical aspects of 14 such polynomials that merge theoretical chemistry and pure mathematics; give examples, differences, and similarities; and relate them to recent research. [ABSTRACT FROM AUTHOR]

Consider G to be a simple graph with n vertices and m edges, and L (G) to be a Laplacian matrix with Laplacian eigenvalues of μ 1 , μ 2 , ... , μ n = z e r o . Write S k (G) = ∑ i = 1 k μ i as the sum of the k-largest Laplacian eigenvalues of G, where k ∈ { 1 , 2 , ... , n } . The motivation of this study is to solve a conjecture in algebraic graph theory for a special type of graph called a wheel graph. Brouwer's conjecture states that S k (G) ≤ m + k + 1 2 , where k = 1 , 2 , ... , n . This paper proves Brouwer's conjecture for wheel graphs. It also provides an upper bound for the sum of the largest Laplacian eigenvalues for the wheel graph W n + 1 , which provides a better approximation for this upper bound using Brouwer's conjecture and the Grone–Merris–Bai inequality. We study the symmetry of wheel graphs and recall an example of the symmetry group of W n + 1 , n ≥ 3 . We obtain our results using majorization methods and illustrate our findings in tables, diagrams, and curves. [ABSTRACT FROM AUTHOR]