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‎Graph coloring is the assignment of one color to each vertex of a graph so that two adjacent vertices are not of the same color‎. ‎The graph coloring problem (GCP) is a matter of combinatorial optimization‎, ‎and the goal of GCP is determining the chromatic number $\chi(G)$‎. ‎Since GCP is an NP-hard problem‎, ‎then in this paper‎, ‎we propose a new approximated algorithm for finding the coloring number (it is an approximation of chromatic number) by using a graph adjacency matrix to colorize or separate a graph‎. ‎To prove the correctness of the proposed algorithm‎, ‎we implement it in MATLAB software‎, ‎and for analysis in terms of solution and execution time‎, ‎we compare our algorithm with some of the best existing algorithms that are already implemented in MATLAB software‎, ‎and we present the results in tables of various graphs‎. ‎Several available algorithms used the largest degree selection strategy‎, ‎while our proposed algorithm uses the graph adjacency matrix to select the vertex that has the smallest degree for coloring‎. ‎We provide some examples to compare the performance of our algorithm to other available methods‎. ‎We make use of the Dolan-Mor\'e performance profiles to assess the performance of the numerical algorithms‎, ‎and demonstrate the efficiency of our proposed approach in comparison with some existing methods‎.

Language phrases are an effective way to express uncertain pieces of information, and easily conforms to the language habits of decision makers to describe the evaluation of things. The consistency judgment of a linguistic judgment matrices is the key to analytic hierarchy process (AHP). If a linguistic judgment matrix has a satisfactory consistency, then the rank of the decision schemes can be determined. In this study, the comparison relation between the decision schemes is first represented by a directed graph. The preference relation matrix of the linguistic judgment matrix is the adjacency matrix of the directed graph. We can use the $ n - 1 $ st power of the preference relation to judge the linguistic judgment matrix whether has a satisfactory consistency. The method is utilized if there is one and only one element in the $ n - 1 $ st power of the preference relation, and the element 1 is not on the main diagonal. Then the linguistic judgment matrix has a satisfactory consistency. If there are illogical judgments, the decision schemes that form a 3-loop can be identified and expressed through the second-order sub-matrix of the preference relation matrix. The feasibility of this theory can be verified through examples. The corresponding schemes for illogical judgments are represented in spatial coordinate system.

Data exploration, usually the first step in data analysis, is a useful method to tackle challenges caused by big geoscience data. It conducts quick analysis of data, investigates the patterns, and generates/refines research questions to guide advanced statistics and machine learning algorithms. The background of this work is the open mineral data provided by several sources, and the focus is different types of associations in mineral properties and occurrences. Researchers in mineralogy have been applying different techniques for exploring such associations. Although the explored associations can lead to new scientific insights that contribute to crystallography, mineralogy, and geochemistry, the exploration process is often daunting due to the wide range and complexity of factors involved. In this study, our purpose is implementing a visualization tool based on the adjacency matrix for a variety of datasets and testing its utility for quick exploration of association patterns in mineral data. Algorithms, software packages, and use cases have been developed to process a variety of mineral data. The results demonstrate the efficiency of adjacency matrix in real-world usage. All the developed works of this study are open source and open access.

Accurate load forecasting, especially in the short term, is crucial for the safe and stable operation of power systems and their market participants. However, as modern power systems become increasingly complex, the challenges of short-term load forecasting are also intensifying. To address this challenge, data-driven deep learning techniques and load aggregation technologies have gradually been introduced into the field of load forecasting. However, data quality issues persist due to various factors such as sensor failures, unstable communication, and susceptibility to network attacks, leading to data gaps. Furthermore, in the domain of aggregated load forecasting, considering the potential interactions among aggregated loads can help market participants engage in cross-market transactions. However, aggregated loads often lack clear geographical locations, making it difficult to predefine graph structures. To address the issue of data quality, this study proposes a model named adversarial graph convolutional imputation network (AGCIN), combined with local and global correlations for imputation. To tackle the problem of the difficulty in predefining graph structures for aggregated loads, this study proposes a learnable adjacency matrix, which generates an adaptive adjacency matrix based on the relationships between different sequences without the need for geographical information. The experimental results demonstrate that the proposed imputation method outperforms other imputation methods in scenarios with random and continuous missing data. Additionally, the prediction accuracy of the proposed method exceeds that of several baseline methods, affirming the effectiveness of our approach in imputation and prediction, ultimately enhancing the accuracy of aggregated load forecasting.

The concept of a swarm of drones assumes the presence of a wireless ad hoc network, in which drones are network nodes and exchanging information with each other. This article is devoted to studying the behavior of an ad hoc network in a transient mode and assessing the characteristics of updating local adjacency matrices (LAM), which allow network nodes to autonomously form packet transmission routes. Using a simulation model, a comparison was made of two multiple access methods to a common data transmission channel in the process of updating matrices: cyclic and random (slotted ALOHA). The simulation results made it possible to determine areas of their effective application: slotted ALOHA is advisable to use with relatively high probabilities of packet distortion (of the order of 0.1 and higher), a numerous nodes (more than 40), and low network connectivity, and cyclic access (CA) is effective at a low level of distortion. It is shown that the completion of the process of updating adjacency matrices (AM) can be judged by such indicators of the exchange of routing information as the achievement of a certain threshold of the number of transmissions and a decrease in the level of network traffic.

Abstract In recent years, air pollution has become increasingly serious and poses a great threat to human health. Timely and accurate air quality prediction is crucial for air pollution early warning and control. Although data-driven air quality prediction methods are promising, there are still challenges in studying spatial–temporal correlations of air pollutants to design effective predictors. To address this issue, a novel model called adaptive adjacency matrix-based graph convolutional recurrent network (AAMGCRN) is proposed in this study. The model inputs Point of Interest (POI) data and meteorological data into a fully connected neural network to learn the weights of the adjacency matrix thereby constructing the self-ringing adjacency matrix and passes the pollutant data with this matrix as input to the Graph Convolutional Network (GCN) unit. Then, the GCN unit is embedded into LSTM units to learn spatio-temporal dependencies. Furthermore, temporal features are extracted using Long Short-Term Memory network (LSTM). Finally, the outputs of these two components are merged and air quality predictions are generated through a hidden layer. To evaluate the performance of the model, we conducted multi-step predictions for the hourly concentration of PM2.5, PM10 and O3 at Fangshan, Tiantan and Dongsi monitoring stations in Beijing. The experimental results show that our method achieves better predicted effects compared with other baseline models based on deep learning. In general, we designed a novel air quality prediction method and effectively addressed the shortcomings of existing studies in learning the spatio-temporal correlations of air pollutants. This method can provide more accurate air quality predictions and is expected to provide support for public health protection and government environmental decision-making.

Multispectral LiDAR can rapidly acquire 3D and spectral information of objects, providing richer features for point cloud semantic segmentation. Despite the remarkable performance of existing graph neural networks in point cloud segmentation, extracting local features still poses challenges in multispectral LiDAR point cloud scenes due to the uneven distribution of geometric and spectral information. To address the prevailing challenges, cutting-edge research predominantly focuses on extracting multiscale local features, compensating for feature extraction shortcomings. Thus, we propose a multiscale adjacency matrix convolutional neural network (MS-AMCNN) for multispectral LiDAR point cloud segmentation. In the MS-AMCNN, a local adjacency matrix convolution module was first proposed to efficiently leverage the point cloud's topological relationships and perceive local geometric features. Subsequently, a multiscale feature extraction architecture was adopted to fuse local geometric features and utilize a global self-attention module to globally model the semantic features of multiscale. The network effectively captures global and local representative features of the point cloud by harnessing the capabilities of convolutional neural networks in local feature modeling and the self-attention mechanism in global semantic feature learning. Experimental results on the Titan dataset demonstrate that the proposed MS-AMCNN network achieves a promising multispectral LiDAR point cloud segmentation performance with an overall accuracy of 94.39% and a mean intersection over union (MIoU) of 86.57%. Compared with other state-of-the-art methods, such as DGCNN, which achieved an MIoU of 85.43%, and RandLA-net, with an MIoU of 85.20%, the proposed approach achieves optimal performance in segmentation.

Accurate load forecasting, especially in the short term, is crucial for the safe and stable operation of power systems and their market participants. However, as modern power systems become increasingly complex, the challenges of short-term load forecasting are also intensifying. To address this challenge, data-driven deep learning techniques and load aggregation technologies have gradually been introduced into the field of load forecasting. However, data quality issues persist due to various factors such as sensor failures, unstable communication, and susceptibility to network attacks, leading to data gaps. Furthermore, in the domain of aggregated load forecasting, considering the potential interactions among aggregated loads can help market participants engage in cross-market transactions. However, aggregated loads often lack clear geographical locations, making it difficult to predefine graph structures. To address the issue of data quality, this study proposes a model named adversarial graph convolutional imputation network (AGCIN), combined with local and global correlations for imputation. To tackle the problem of the difficulty in predefining graph structures for aggregated loads, this study proposes a learnable adjacency matrix, which generates an adaptive adjacency matrix based on the relationships between different sequences without the need for geographical information. The experimental results demonstrate that the proposed imputation method outperforms other imputation methods in scenarios with random and continuous missing data. Additionally, the prediction accuracy of the proposed method exceeds that of several baseline methods, affirming the effectiveness of our approach in imputation and prediction, ultimately enhancing the accuracy of aggregated load forecasting. [ABSTRACT FROM AUTHOR]

This article focused on the notion of the non-abelian tensor product of groups of prime power order. Particularly, it presented new graph named as Non-compatible action graph and discussed some of its properties. Moreover, this graph concentrated on the case of non-compatible actions of the tensor product of two finite -groups. Furthermore, its adjacency matrix has been determined and discussed in detail. Moreover, the adjacency matrix has been denoted by A(G) and its inputs are 1 whenever there is adjacency and 0 otherwise.

Image clustering is a research hotspot in machine learning and computer vision. Existing graph-based semi-supervised deep clustering methods suffer from three problems: 1) because clustering uses only high-level features, the detailed information contained in shallow-level features is ignored; 2) most feature extraction networks employ the step odd convolutional kernel, which results in an uneven distribution of receptive field intensity; and 3) because the adjacency matrix is precomputed and fixed, it cannot adapt to changes in the relationship between samples. To solve the above problems, we propose a novel graph-based semi-supervised deep clustering method for image clustering. First, the parity cross-convolutional feature extraction and fusion module is used to extract high-quality image features. Then, the clustering constraint layer is designed to improve the clustering efficiency. And, the output layer is customized to achieve unsupervised regularization training. Finally, the adjacency matrix is inferred by actual network prediction. A graph-based regularization method is adopted for unsupervised training networks. Experimental results show that our method significantly outperforms state-of-the-art methods on USPS, MNIST, street view house numbers (SVHN), and fashion MNIST (FMNIST) datasets in terms of ACC, normalized mutual information (NMI), and ARI.

In recent years, air pollution has become increasingly serious and poses a great threat to human health. Timely and accurate air quality prediction is crucial for air pollution early warning and control. Although data-driven air quality prediction methods are promising, there are still challenges in studying spatial-temporal correlations of air pollutants to design effective predictors. To address this issue, a novel model called adaptive adjacency matrix-based graph convolutional recurrent network (AAMGCRN) is proposed in this study. The model inputs Point of Interest (POI) data and meteorological data into a fully connected neural network to learn the weights of the adjacency matrix thereby constructing the self-ringing adjacency matrix and passes the pollutant data with this matrix as input to the Graph Convolutional Network (GCN) unit. Then, the GCN unit is embedded into LSTM units to learn spatio-temporal dependencies. Furthermore, temporal features are extracted using Long Short-Term Memory network (LSTM). Finally, the outputs of these two components are merged and air quality predictions are generated through a hidden layer. To evaluate the performance of the model, we conducted multi-step predictions for the hourly concentration of PM 2.5 , PM 10 and O 3 at Fangshan, Tiantan and Dongsi monitoring stations in Beijing. The experimental results show that our method achieves better predicted effects compared with other baseline models based on deep learning. In general, we designed a novel air quality prediction method and effectively addressed the shortcomings of existing studies in learning the spatio-temporal correlations of air pollutants. This method can provide more accurate air quality predictions and is expected to provide support for public health protection and government environmental decision-making., (© 2024. The Author(s).)

Graph coloring is the assignment of one color to each vertex of a graph so that two adjacent vertices are not of the same color. The graph coloring problem (GCP) is a matter of combinatorial optimization, and the goal of GCP is determining the chromatic number (G). Since GCP is an NP-hard problem, then in this paper, we propose a new approximated algorithm for finding the coloring number (it is an approximation of chromatic number) by using a graph adjacency matrix to colorize or separate a graph. To prove the correctness of the proposed algorithm, we implement it in MATLAB software, and for analysis in terms of solution and execution time, we compare our algorithm with some of the best existing algorithms that are already implemented in MATLAB software, and we present the results in tables of various graphs. Several available algorithms used the largest degree selection strategy, while our proposed algorithm uses the graph adjacency matrix to select the vertex that has the smallest degree for coloring. We provide some examples to compare the performance of our algorithm to other available methods. We make use of the Dolan-Moré performance profiles to assess the performance of the numerical algorithms, and demonstrate the efficiency of our proposed approach in comparison with some existing methods. [ABSTRACT FROM AUTHOR]

Let GG be a graph. Then, the inverse graph G−1{G}^{-1} of GG is defined to be a graph that has adjacency matrix similar to the inverse of the adjacency matrix of GG, where the similarity matrix is ±1\pm 1 diagonal matrix. In this article, we introduced a generalization of this definition that serves the mixed graphs where the definition applied for the α\alpha -Hermitian adjacency matrices of mixed graphs. Furthermore, for a class of unicyclic graphs, we were able to find an inverse mixed graph for a graph GG, where it was proven that G−1{G}^{-1} does not exist.

In this article, we study eigenvalues λ\lambda and their associated eigenvectors xx of the adjacency matrices AA of balanced signed graphs. Balanced signed graphs were first introduced and studied by Harary to handle a problem in social psychology. Harary showed in 1953 that a signed graph is balanced if and only if its vertex set VV can be divided into two sets (either of which may be empty), XX and YY, so that each edge between the sets is negative and each within a set is positive. Based on this fundamental theorem for the balanced signed graphs, vertices of a balanced signed graph can be labeled in a way so that its adjacency matrix is well structured. Using this special structure, we find algebraically all eigenvalues and their associated eigenvectors of the adjacency matrix AA of a given balanced signed graph. We present in this study eigenpairs (λ,x)\left(\lambda ,x) of adjacency matrices of balanced signed graphs with some special structures.

Counting perfect matchings is an interesting and challenging combinatorial task. It has important applications in statistical physics and chemistry. As the general problem is #P-complete, it is usually tackled by randomized heuristics and approximation schemes. Let GG and HH be two graphs. Denote by G×HG\times H the Cartesian product of graphs GG and HH. The number of perfect matchings of some types of Cartesian product G×HG\times H is investigated by the Pfaffian method, where G≅P2G\cong {P}_{2}, P3{P}_{3}, P4{P}_{4} or C4{C}_{4}, and H≅TH\cong T or HH is a non-bipartite graph with a unique cycle. In this article, we construct a Pfaffian orientation of the graph G×P2G\times {P}_{2}, where GG is a non-bipartite graph with kk odd cycles. We obtain the counting formula on the number of perfect matchings of G×P2G\times {P}_{2}.

In this article, we study a question of Hong from 1993 related to the minimum spectral radii of the adjacency matrices of connected graphs of given order and size. Hong asked if it is true that among all connected graphs of given number of vertices nn and number of edges ee, the graphs having minimum spectral radius (the minimizer graphs) must be almost regular, meaning that the difference between their maximum degree and their minimum degree is at most one. In this article, we answer Hong’s question positively for various values of nn and ee, and in several cases, we determined the graphs with minimum spectral radius.

In this paper, we introduce the subset-strong product of graphs and give a method for calculating the adjacency spectrum of this product. In addition, exact expressions for the first and second Zagreb indices of the subset-strong products of two graphs are reported. Examples are provided to illustrate the applications of this product in some growing graphs and complex networks.

In today's world, various approaches and parameters exist for designing a plan and determining its spatial, placement. Hence, various modes for identifying crucial locations can be explored when an architectural plan is designed in different dimensions. While designing all these modes takes considerable time, there are numerous potential applications for artificial intelligence (AI) in this domain. This study aims to compute and use an adjacency matrix to generate architectural residential plans. Additionally, it develops a plan generation algorithm in Rhinoceros software, utilizing the Grasshopper plugin to create a dataset of architectural plans. In the following step, the data was entered into a neural network to identify the architectural plan's type, furniture, icons, and use of spaces, which was achieved using YOLOv4, EfficientDet, YOLOv5, DetectoRS, and RetinaNet. The algorithm's execution, testing, and training were conducted using Darknet and PyTorch. The research dataset comprises 12,000 plans, with 70% employed in the training phase and 30% in the testing phase. The network was appropriately trained practically and precisely in relation to an average precision (AP) resulting of 91.50%. After detecting the types of space use, the main research algorithm has been designed and coded, which includes determining the adjacency matrix of architectural plan spaces in seven stages. All research processes were conducted in Python, including dataset preparation, network object detection, and adjacency matrix algorithm design. Finally, the adjacency matrix is given to the input of the proposed plan generator network, which consequently, based on the resulting adjacency, obtains different placement modes for spaces and furniture.

Buildings are responsible for around 30% and 42% of the consumed energy at the global and European levels, respectively. Accurate building power consumption estimation is crucial for resource saving. This research investigates the combination of graph convolutional networks (GCNs) and long short-term memory networks (LSTMs) to analyze power building consumption, thereby focusing on predictive modeling. Specifically, by structuring graphs based on Pearson’s correlation and Euclidean distance methods, GCNs are employed to discern intricate spatial dependencies, and LSTM is used for temporal dependencies. The proposed models are applied to data from a multistory, multizone educational building, and they are then compared with baseline machine learning, deep learning, and statistical models. The performance of all models is evaluated using metrics such as the mean absolute error (MAE), mean squared error (MSE), R-squared (R2), and the coefficient of variation of the root mean squared error (CV(RMSE)). Among the proposed computation models, one of the Euclidean-based models consistently achieved the lowest MAE and MSE values, thus indicating superior prediction accuracy. The suggested methods seem promising and highlight the effectiveness of GCNs in improving accuracy and reliability in predicting power consumption. The results could be useful in the planning of building energy policies by engineers, as well as in the evaluation of the energy management of structures.

The prediction for Multivariate Time Series (MTS) explores the interrelationships among variables at historical moments, extracts their relevant characteristics, and is widely used in finance, weather, complex industries and other fields. Furthermore, it is important to construct a digital twin system. However, existing methods do not take full advantage of the potential properties of variables, which results in poor predicted accuracy. In this paper, we propose the Adaptive Fused Spatial-Temporal Graph Convolutional Network (AFSTGCN). First, to address the problem of the unknown spatial-temporal structure, we construct the Adaptive Fused Spatial-Temporal Graph (AFSTG) layer. Specifically, we fuse the spatial-temporal graph based on the interrelationship of spatial graphs. Simultaneously, we construct the adaptive adjacency matrix of the spatial-temporal graph using node embedding methods. Subsequently, to overcome the insufficient extraction of disordered correlation features, we construct the Adaptive Fused Spatial-Temporal Graph Convolutional (AFSTGC) module. The module forces the reordering of disordered temporal, spatial and spatial-temporal dependencies into rule-like data. AFSTGCN dynamically and synchronously acquires potential temporal, spatial and spatial-temporal correlations, thereby fully extracting rich hierarchical feature information to enhance the predicted accuracy. Experiments on different types of MTS datasets demonstrate that the model achieves state-of-the-art single-step and multi-step performance compared with eight other deep learning models.

AbstractThe aim of this paper is to study the determinant and inverse of the adjacency matrices of weighted and directed versions of Boolean graphs. Our approach is recursive. We describe the adjacency matrix of a weighted Boolean graph in terms of the adjacency matrix of a smaller-sized weighted Boolean graph. This allows us to compute the determinant and inverse of the adjacency matrix of a weighted Boolean graph recursively. In particular, we show that the determinant of a directed Boolean graph is 1. Further, using a classical theorem of Cayley which expresses the determinant of any skew-symmetric matrix as a square of its Pfaffian, we show that for any directed Boolean graph, the characteristic polynomial has all its even degree coefficients strictly positive with the odd ones being zero.

Human pose estimation is a hot research in the field of computer vision. Existing algorithms perform well in normal scenarios but struggle in occlusion environments. In order to address occlusion problems, we firstly design a multi-scale feature reinforcement backbone (MFRI) to obtain accurate features, which contains three sub-modules. The first sub-module reinforces detailed keypoint features and eliminates redundant background, The second sub-module assigns different weights to different scale information based on importance. The fourth sub-module integrates different scale feature information using a cascaded method. In addition, we design an occlusion handling method. To begin with, we design a multi-scale feature extractor (MSFE) to construct human topological feature, and propose a feature similarity matching method between keypoint detail features and topological features to identify occluded keypoints and eliminate obstacle features. Based on similarity, we design a feature compensation method to extract common attributes among all keypoints and unique features of each keypoint from multiple aspects to compensate for occluded features. Finally, we propose an adjacency matrix improvement method to enhance its ability for describing node relationships. We conduct comparative experiments on the COCO2017 dataset, COCO-Wholebody dataset, and CrowdPose dataset, achieving accuracy of 78.8%, 66.3%, and 77.8%, respectively. Additionally, we design a series of ablation experiments and visualizations. The results show that our method has better performance in handling occlusions.

Video motion recognition plays a crucial role in advanced sports analysis. With video motion recognition, sports analytics has become more data-driven and result-oriented, significantly enhancing the professionalism and efficiency in the sports domain. Over the years, the accuracy of skelecton-based motion recognition algorithms using Recurrent Neural Networks (RNNs) and Convolutional Neural Networks (ConvNets) has plateaued on relevant datasets, struggling to achieve further breakthroughs. This is partly because RNNs lack sufficient capability to model spatial structural features, and while ConvNets can alleviate difficulties in modeling spatial structures, the conversion of skelecton sequences into RGB pseudo-images inherently leads to some information loss. Moreover, ConvNets are not particularly adept at modeling temporal features. Since the associations between articulations in bodily skelectons are better represented using a graph structure, Graph Convolution Network (GCN)-based skelecton motion recognition methods have gained more attention. Recent advancements in Shift Graph Convolutional Network (S-GCN) have enhanced the expressiveness of spatial graphs while improving the versatility of spatial and temporal graph receptive fields. To further enhance this versatility, we propose the Dynamic Motion-Aware Shift Graph Convolutional Network (DMA-SGCN) for video motion recognition. Specifically, we introduce a data-aware driven method to represent associations between articulations. By analyzing the attributes of different motions and combining them with the natural associations of the bodily skelecton, we compute articulation affinities through representation learning. This approach not only improves the accuracy in defining articulation associations but also enhances the awareness of related articulation associations during motion behaviours. Furthermore, we dynamically use the adjacency matrix of skeletal data to guide the feature transfer between articulations. This topological method allows for more effective shift transformations based on articulation associations, addressing the issue of rigid receptive fields in previous GCNs for motion recognition. Contrast and ablation experiments on the largest 3D motion recognition dataset demonstrate that starting from the skeletal data and the motions themselves enables more accurate excavation of dynamic associations of skeletal articulations in the individual motion recognition.

In addressing the challenges of trajectory prediction in multi-ship interaction scenarios and aiming to improve the accuracy of multi-ship trajectory prediction, this paper proposes a multi-ship trajectory prediction model, GL-STGCNN. The GL-STGCNN model employs a ship interaction adjacency matrix extraction module to obtain a more reasonable ship interaction adjacency matrix. Additionally, after obtaining the distribution of predicted trajectories using the model, a model predictive control trajectory correction method is introduced to enhance the accuracy and reasonability of the predicted trajectories. Through quantitative analysis of different datasets, it was observed that GL-STGCNN outperforms previous prediction models with a 31.8% improvement in the average displacement error metric and a 16.8% improvement in the final displacement error metric. Furthermore, trajectory correction through model predictive control shows a performance boost of 44.5% based on the initial predicted trajectory distribution. While GL-STGCNN excels in multi-ship interaction trajectory prediction by reasonably modeling ship interaction adjacency matrices and employing trajectory correction, its performance may vary in different datasets and ship motion patterns. Future work could focus on adapting the model’s ship interaction adjacency matrix modeling to diverse environmental scenarios for enhanced performance.

In this note we show that for each positive integer a⩾2 there exist infinitely many trees whose spectral radius is equal to 2a. Such trees are obtained by replacing the central edge of the double star S(a,2a−2) with suitable bidegreed caterpillars.

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.

Let Gσ be an oriented graph with underlying simple graph G. The skew-adjacency matrix of Gσ is the {0, 1,-1}-matrix S = S(Gσ) = [sij ], such that sij = 1 if (vi, vj) is an arc in Gσ, sij = -1 if (vj, vi) is an arc in Gσ and sij = 0, otherwise. In this paper, all connected oriented graphs with three distinct skew-eigenvalues 0 and ±2i are characterized. [ABSTRACT FROM AUTHOR]

The demolition issue of long-span spatial lattice structures has become increasingly prominent with the rapid urban development and building upgrades. Building deconstruction offers superior economic and environmental benefits compared to traditional demolition and therefore should be further promoted and applied. Dismantling in blocks is an efficient approach for deconstruction of long-span spatial lattice structures, surpassing the practice of dismantling individual components one by one. However, dividing the entire structure into substructures considering the convenience and safety of dismantling operations is the first challenge of dismantling in blocks yet to be overcome. In this study, a block planning method was proposed to aid practitioners in making high-quality block schemes. The topological relation of spatial grids defined by the adjacency matrix is the basis. The comprehensive evaluation index was established reflecting the operability, efficiency, and safety of the block scheme. Taking the combination of spatial grids as the variable and the comprehensive evaluation index combined with constraint functions as the objective, the block planning was formulated as an optimization problem. The genetic algorithm was adopted to solve this problem, and two examples were presented for verification. Results confirmed that the proposed block planning method is effective to generate the high-quality block scheme that meets topology, quantity, and lifting weight requirements and has great synthetical performance. The consideration of structural symmetry can accelerate the optimization process. The zoning planning strategy provides a tool for practitioners to make the block scheme that meets the lifting weight requirements of different zones when the crane's lifting capacity is limited by operating space. The proposed block planning method will enrich the construction management technologies of dismantling projects and help engineers make efficient and wise decisions. [ABSTRACT FROM AUTHOR]

In this paper, for the visual image shaping of folk culture elements in city branding, the intelligent extraction technology of digital image with image pre-processing, color space conversion, and color extraction as the three core modules is proposed. In terms of image preprocessing, spatial domain filtering is used as a means to explore the two main algorithms of linear filtering and nonlinear filtering, and in the color space conversion module, the RGB color space is converted to the CIE Lab color space with the help of CIE XYZ color. The color map extraction module uses Python language to construct the color collocation relationship network using K-means and color adjacency matrix visualization. Finally, the quantitative experiment on the effect of the city brand visual image that integrates the elements of folk culture is carried out, and 65 city brand visual image samples are quantitatively analyzed, while the AMOS model is applied to test the city brand visual image.

In the paper, we introduce the signless Laplacian ABC-matrix Q̃(G)=D¯(G)+Ã(G), where D¯(G) is the diagonal matrix of ABC-degrees and Ã(G) is the ABC-matrix of G. The eigenvalues of the matrix Q̃(G) are the signless Laplacian ABC-eigenvalues of G. We give some basic properties of the matrix Q̃(G), which includes relating independence number and clique number with signless Laplacian ABC-eigenvalues. For bipartite graphs, we show that the signless Laplacian ABC-spectrum and the Laplacian ABC-spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian ABC-eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian ABC-eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q̃(G)−tr(Q̃(G))nI, called the signless Laplacian ABC-energy of G. We obtain some upper and lower bounds for signless Laplacian ABC-energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the ABC-energy with the signless Laplacian ABC-energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian ABC-energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same ABC-energy.

In this article, we discuss the spectral properties of the general extended adjacency matrix for chain graphs. In particular, we discuss the eigenvalues of the general extended adjacency matrix of the chain graphs and obtain its general extended adjacency inertia. We obtain bounds for the largest and the smallest general extended adjacency eigenvalues and characterize the extremal graphs. We also obtain a lower bound for the spread of the general extended adjacency matrix. We characterize chain graphs with all the general extended adjacency eigenvalues being simple and chain graphs that are non-singular under the general extended adjacency matrix. Further, we determine the explicit formula for the determinant and the trace of the square of the general extended adjacency matrix of chain graphs. Finally, we discuss the energy of the general extended adjacency matrix and obtain some bounds for it. We characterize the extremal chain graphs attaining these bounds.

Campus safety is related to the healthy growth of students, and the research focus of this paper is to explore the impact of Internet of Things (IoT) technology on campus safety prevention and control. Based on the correlation between ISM and DEMATEL methods, this paper constitutes an integrated DEMATEL-ISM model by using neighbor matrix and reachable matrix to transform each other. It constructs a risk factor model for campus security prevention and control evaluation based on Internet of Things technology according to the modeling steps after soliciting experts’ suggestions through questionnaires. Combined with the SEM model to estimate students’ campus risk identification ability, the path index was used to construct the college security awareness score index to explore the relationship between latent variables (influencing factors) and observed variables (risk identification ability). The results show that every 1% increase in the safety climate of the social environment will lead to the rise of 0.58%, 0.20%, and 0.52% in the on-campus environment factor, the campus safety management team building factor, and the students’ literacy factor, respectively. A 0.17% increase in the on-campus environment will occur with every 1% increase in students’ literacy in higher education. This study contributes to a deeper understanding of the role of IoT technology in enhancing campus security prevention and control. It can provide deep research references for subsequent risk grading evaluation, risk warning prevention and control, and supervision in campus risk management.

This study employs deep learning techniques to present a compelling approach for modeling brain connectivity in EEG motor imagery classification through graph embedding. The compelling aspect of this study lies in its combination of graph embedding, deep learning, and different brain connectivity types, which not only enhances classification accuracy but also enriches the understanding of brain function. The approach yields high accuracy, providing valuable insights into brain connections and has potential applications in understanding neurological conditions. The proposed models consist of two distinct graph-based convolutional neural networks, each leveraging different types of brain connectivities to enhance classification performance and gain a deeper understanding of brain connections. The first model, Adjacency-based Convolutional Neural Network Model (Adj-CNNM), utilizes a graph representation based on structural brain connectivity to embed spatial information, distinguishing it from prior spatial filtering approaches dependent on subjects and tasks. Extensive tests on a benchmark dataset-IV-2a demonstrate that an accuracy of 72.77% is achieved by the Adj-CNNM, surpassing baseline and state-of-the-art methods. The second model, Phase Locking Value Convolutional Neural Network Model (PLV-CNNM), incorporates functional connectivity to overcome structural connectivity limitations and identifies connections between distinct brain regions. The PLV-CNNM achieves an overall accuracy of 75.10% across the 1–51 Hz frequency range. In the preferred 8–30 Hz frequency band, known for motor imagery data classification (including α, μ, and β waves), individual accuracies of 91.9%, 90.2%, and 85.8% are attained for α, μ, and β, respectively. Moreover, the model performs admirably with 84.3% accuracy when considering the entire 8–30 Hz band. Notably, the PLV-CNNM reveals robust connections between different brain regions during motor imagery tasks, including the frontal and central cortex and the central and parietal cortex. These findings provide valuable insights into brain connectivity patterns, enriching the comprehension of brain function. Additionally, the study offers a comprehensive comparative analysis of diverse brain connectivity modeling methods.

Accurately predicting the active power output of offshore wind power is of great significance for reducing the uncertainty in new power systems. By utilizing the spatiotemporal correlation characteristics among wind turbine unit outputs, this paper embeds the Diffusion Convolutional Neural Network (DCNN) into the Gated Recurrent Unit (GRU) for the feature extraction of spatiotemporal correlations in wind turbine unit outputs. It also combines graph structure learning to propose a sequence-to-sequence model for ultra-short-term power prediction in large offshore wind farms. Firstly, the electrical connection graph within the wind farm is used to preliminarily determine the reference adjacency matrix for the wind turbine units within the farm, injecting prior knowledge of the adjacency matrix into the model. Secondly, a convolutional neural network is utilized to convolve the historical curves of units within the farm along the time dimension, outputting a unit connection probability vector. The Gumbel–softmax reparameterization method is then used to make the probability vector differentiable, thereby generating an optimal adjacency matrix for the prediction task based on the probability vector. At the same time, the difference between the two adjacency matrices is added as a regularization term to the loss function to reduce model overfitting. The simulation of actual cases shows that the proposed model has good predictive performance in ultra-short-term power prediction for large offshore wind farms. [ABSTRACT FROM AUTHOR]

We extend the notion of balance from the realm of signed and gain graphs to conjugate skew gain graphs which are skew gain graphs where the labels on the oriented edges get conjugated when we reverse the orientation. We characterize the balance in a conjugate skew gain graph in several ways especially by dealing with its adjacency matrix and the g -Laplacian matrix. We also deal with the concept of anti-balance in conjugate skew gain graphs. [ABSTRACT FROM AUTHOR]

Emotion is one of the most important higher cognitive functions of the human brain and plays an important role in transaction processing and decisions. In traditional emotion recognition studies, the frequency band features in EEG signals have been shown to have a high correlation with emotion production. However, traditional emotion recognition methods cannot satisfactorily solve the problem of individual differences in subjects and data heterogeneity in EEG, and subject-independent emotion recognition based on EEG signals has attracted extensive attention from researchers. In this paper, we propose a subject-independent emotion recognition model based on adaptive extraction of layer structure based on frequency bands (BFE-Net), which is adaptive in extracting EEG map features through the multi-graphic layer construction module to obtain a frequency band-based multi-graphic layer emotion representation. To evaluate the performance of the model in subject-independent emotion recognition studies, extensive experiments are conducted on two public datasets including SEED and SEED-IV. The experimental results show that in most experimental settings, our model has a more advanced performance than the existing studies of the same type. In addition, the visualization of brain connectivity patterns reveals that some of the findings are consistent with previous neuroscientific validations, further validating the model in subject-independent emotion recognition studies.

With the rapid advancement of deep learning, 3D human pose estimation has largely freed itself from reliance on manually annotated methods. The effective utilization of joint features has become significant. Utilizing 2D human joint information to predict 3D human skeletons is of paramount importance. Effectively leveraging 2D joint data can improve the accuracy of 3D human skeleton prediction. In this paper, we propose the SCGFormer model to reduce the error in predicting human skeletal poses in three-dimensional space. The network architecture of SCGFormer encompasses Transformer and two distinct types of graph convolution, organized into two interconnected modules: SGraAttention and AcChebGconv. SGraAttention extracts global feature information from each 2D human joint, thereby augmenting local feature learning by integrating prior knowledge of human joint relationships. Simultaneously, AcChebGconv broadens the receptive field for graph structure information and constructs implicit joint relationships to aggregate more valuable adjacent features. SCGraFormer is tested on widely recognized benchmark datasets such as Human3.6M and MPI-INF-3DHP and achieves excellent results. In particular, on Human3.6M, our method achieves the best results in 9 actions (out of a total of 15 actions), with an overall average error reduction of about 1.5 points compared to state-of-the-art methods, demonstrating the excellent performance of SCGFormer.

In this article, we discuss the spectral properties of the general extended adjacency matrix for chain graphs. In particular, we discuss the eigenvalues of the general extended adjacency matrix of the chain graphs and obtain its general extended adjacency inertia. We obtain bounds for the largest and the smallest general extended adjacency eigenvalues and characterize the extremal graphs. We also obtain a lower bound for the spread of the general extended adjacency matrix. We characterize chain graphs with all the general extended adjacency eigenvalues being simple and chain graphs that are non-singular under the general extended adjacency matrix. Further, we determine the explicit formula for the determinant and the trace of the square of the general extended adjacency matrix of chain graphs. Finally, we discuss the energy of the general extended adjacency matrix and obtain some bounds for it. We characterize the extremal chain graphs attaining these bounds. [ABSTRACT FROM AUTHOR]

Precisely segmenting the organs at risk (OARs) in computed tomography (CT) plays an important role in radiotherapy's treatment planning, aiding in the protection of critical tissues during irradiation. Renowned deep convolutional neural networks (DCNNs) and prevailing transformer-based architectures are widely utilized to accomplish the segmentation task, showcasing advantages in capturing local and contextual characteristics. Graph convolutional networks (GCNs) are another specialized model designed for processing the nongrid dataset, e.g., citation relationship. The DCNNs and GCNs are considered as two distinct models applicable to the grid and nongrid datasets, respectively. Motivated by the recently developed dynamic-channel GCN (DCGCN) that attempts to leverage the graph structure to enhance the feature extracted by the DCNNs, this paper proposes a novel architecture termed adaptive sparse GCN (ASGCN) to mitigate the inherent limitations in DCGCN from the aspect of node's representation and adjacency matrix's construction. For the node's representation, the global average pooling used in the DCGCN is replaced by the learning mechanism to accommodate the segmentation task. For the adjacency matrix, an adaptive regularization strategy is leveraged to penalize the coefficient in the adjacency matrix, resulting in a sparse one that can better exploit the relationships between nodes. Rigorous experiments on multiple OARs' segmentation tasks of the head and neck demonstrate that the proposed ASGCN can effectively improve the segmentation accuracy. Comparison between the proposed method and other prevalent architectures further confirms the superiority of the ASGCN. [ABSTRACT FROM AUTHOR]

Let A(G) be the adjacency matrix of a simple connected undirected graph G. A graph G of order n is said to be non-singular (respectively singular) if A(G) is non-singular (respectively singular). The spectrum of a graph G is the set of all its eigenvalues denoted by spec(G). The antireciprocal (respectively reciprocal) eigenvalue property for a graph G can be defined as "Let G be a non-singular graph G if the negative reciprocal (respectively positive reciprocal) of each eigenvalue is likewise an eigenvalue of G, then G has anti-reciprocal (respectively reciprocal) eigenvalue property." Furthermore, a graph G is said to have strong anti-reciprocal eigenvalue property (resp. strong reciprocal eigenvalue property) if the eigenvalues and their negative (resp. positive) reciprocals are of same multiplicities. In this article, graphs satisfying anti-reciprocal eigenvalue (or property (-R)) and strong anti-reciprocal eigenvalue property (or property (-SR)) are discussed. [ABSTRACT FROM AUTHOR]

In addressing the challenges of trajectory prediction in multi-ship interaction scenarios and aiming to improve the accuracy of multi-ship trajectory prediction, this paper proposes a multi-ship trajectory prediction model, GL-STGCNN. The GL-STGCNN model employs a ship interaction adjacency matrix extraction module to obtain a more reasonable ship interaction adjacency matrix. Additionally, after obtaining the distribution of predicted trajectories using the model, a model predictive control trajectory correction method is introduced to enhance the accuracy and reasonability of the predicted trajectories. Through quantitative analysis of different datasets, it was observed that GL-STGCNN outperforms previous prediction models with a 31.8% improvement in the average displacement error metric and a 16.8% improvement in the final displacement error metric. Furthermore, trajectory correction through model predictive control shows a performance boost of 44.5% based on the initial predicted trajectory distribution. While GL-STGCNN excels in multi-ship interaction trajectory prediction by reasonably modeling ship interaction adjacency matrices and employing trajectory correction, its performance may vary in different datasets and ship motion patterns. Future work could focus on adapting the model's ship interaction adjacency matrix modeling to diverse environmental scenarios for enhanced performance. [ABSTRACT FROM AUTHOR]

In the paper, we introduce the signless Laplacian A B C -matrix Q ̃ (G) = D ¯ (G) + A ̃ (G) , where D ¯ (G) is the diagonal matrix of A B C -degrees and A ̃ (G) is the A B C -matrix of G. The eigenvalues of the matrix Q ̃ (G) are the signless Laplacian A B C -eigenvalues of G. We give some basic properties of the matrix Q ̃ (G) , which includes relating independence number and clique number with signless Laplacian A B C -eigenvalues. For bipartite graphs, we show that the signless Laplacian A B C -spectrum and the Laplacian A B C -spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian A B C -eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian A B C -eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q ̃ (G) − t r (Q ̃ (G)) n I , called the signless Laplacian A B C -energy of G. We obtain some upper and lower bounds for signless Laplacian A B C -energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the A B C -energy with the signless Laplacian A B C -energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian A B C -energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same A B C -energy. [ABSTRACT FROM AUTHOR]

The purpose of this study is to develop an evaluation method for assessing the relative relationship between gear tooth shape deviations on every gear tooth using network theory. Our previous study introduced a method for representing the phase difference between each helix deviation as a network, demonstrating that it is possible to identify the relative relationship between gear tooth deviations. However, there has been no in-depth analysis of the impact of injection molding on the gear's phase difference network. In this paper, we begin by measuring the gear tooth shape deviation, calculating the correlation coefficient, and expressing it as an adjacency matrix. When the adjacency matrix was visualized and displayed as a pixel plot, a periodic pattern was formed. The relationship between the position of the gate used to inject molten resin material into the injection mold and the pixel plot were then investigated in detail, and it was confirmed that the helix deviation network of a gear manufactured with the injection molding process is useful as a new indicator for the manufacturing error of injection-molded plastic gear. [ABSTRACT FROM AUTHOR]

We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix called an "antisymmetrized adjacency matrix", specifically tailored for cycle digraphs (T 1 lattice) and simple directed paths (B 1 lattice). The nontrivial relationship between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the antisymmetrized adjacency matrix, along with its extensions to higher dimensions, is equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D -dimensional graph results from a Cartesian product of cycle digraphs (T 1 lattice) and simple directed paths (B 1 lattice). [ABSTRACT FROM AUTHOR]

Predicting olfactory perceptions from odorant molecules is challenging due to the complex and potentially discontinuous nature of the perceptual space for smells. In this study, we introduce a deep learning model, Mol-PECO (Molecular Representation by Positional Encoding of Coulomb Matrix), designed to predict olfactory perceptions based on molecular structures and electrostatics. Mol-PECO learns the efficient embedding of molecules by utilizing the Coulomb matrix, which encodes atomic coordinates and charges, as an alternative of the adjacency matrix and its Laplacian eigenfunctions as positional encoding of atoms. With a comprehensive dataset of odor molecules and descriptors, Mol-PECO outperforms traditional machine learning methods using molecular fingerprints and graph neural networks based on adjacency matrices. The learned embeddings by Mol-PECO effectively capture the odor space, enabling global clustering of descriptors and local retrieval of similar odorants. This work contributes to a deeper understanding of the olfactory sense and its mechanisms. [ABSTRACT FROM AUTHOR]