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The relation is a semantic expression relevant to two named entities in a sentence. Since a sentence usually contains several named entities, it is essential to learn a structured sentence representation that encodes dependency information specific to the two named entities. In related work, graph convolutional neural networks are widely adopted to learn semantic dependencies, where a dependency tree initializes the adjacency matrix. However, this approach has two main issues. First, parsing a sentence heavily relies on external toolkits, which can be error-prone. Second, the dependency tree only encodes the syntactical structure of a sentence, which may not align with the relational semantic expression. In this paper, we propose an automatic graph learning method to autonomously learn a sentence's structural information. Instead of using a fixed adjacency matrix initialized by a dependency tree, we introduce an Adaptive Adjacency Matrix to encode the semantic dependency between tokens. The elements of this matrix are dynamically learned during the training process and optimized by task-relevant learning objectives, enabling the construction of task-relevant semantic dependencies within a sentence. Our model demonstrates superior performance on the TACRED and SemEval 2010 datasets, surpassing previous works by 1.3% and 0.8%, respectively. These experimental results show that our model excels in the relation extraction task, outperforming prior models. [ABSTRACT FROM AUTHOR]

Video summarization aims to select key frames or key shots to create summaries for fast retrieval, compression, and efficient browsing of videos. Graph neural networks efficiently capture information about graph nodes and their neighbors, but ignore the dynamic dependencies between nodes. To address this challenge, we propose an innovative Adaptive Graph Convolutional Adjacency Matrix Network (TAMGCN), leveraging the attention mechanism to dynamically adjust dependencies between graph nodes. Specifically, we first segment shots and extract features of each frame, then compute the representative features of each shot. Subsequently, we utilize the attention mechanism to dynamically adjust the adjacency matrix of the graph convolutional network to better capture the dynamic dependencies between graph nodes. Finally, we fuse temporal features extracted by Bi-directional Long Short-Term Memory network with structural features extracted by the graph convolutional network to generate high-quality summaries. Extensive experiments are conducted on two benchmark datasets, TVSum and SumMe, yielding F1-scores of 60.8% and 53.2%, respectively. Experimental results demonstrate that our method outperforms most state-of-the-art video summarization techniques. [ABSTRACT FROM AUTHOR]

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]

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]

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 1967, Rosa introduced β-labeling which was then popularized by Golomb under the name graceful. Graceful labeling on a graph G is an injective function f:V(G)→{0, 1, 2, ..., |E(G)|} such that, when each edge uv∈E(G) is assigned the label |f(u)-f(v)| the resulting edge labels are distinct. If graph G has graceful labeling then G is called a graceful graph. Rosa also introduced α-labeling on graph G which is a graceful labeling f with an additional condition that there is λ∈{1, 2, ..., |E(G)|} so that for every edge uv∈E(G) where f(u)

With the deterioration growth of water pipes, the frequency of identifying segments and unintended isolation in water distribution networks is expected to increase, because it can help identify the number of nodes that cannot be supplied and their locations when a failure occurs. Therefore, a methodology for identifying segments and unintended isolation was developed and verified using examples. In addition, a case study was conducted to confirm that the proposed methodology could be applied to actual fields. The proposed method only requires the node-based adjacency matrix K for identification, where the connectivity of the current network can be intuitively understood with the constructed matrices. Moreover, this methodology is easy to adopt using the single row-first search method. The proposed framework is compatible with the data storage format of hydraulic analysis software. Finally, the proposed methodology can be used in various research on configuring and analyzing water distribution networks. [ABSTRACT FROM AUTHOR]

The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest in this question dates back to the early 1980 s, when P. J. Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable adjacency matrix. This was answered in the affirmative by Babai (J Graph Theory 9:363–370, 1985). Then Babai posed the open problems of constructing a 2-arc-transitive digraph and a vertex-primitive digraph whose adjacency matrices are not diagonalizable. In this paper, we solve Babai's problems by constructing an infinite family of s-arc-transitive digraphs for each integer s ≥ 2 , and an infinite family of vertex-primitive digraphs, both of whose adjacency matrices are non-diagonalizable. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In the spectral clustering methods, different from the network division based on edges, some research has begun to divide the network based on network motifs; the corresponding objective function of partition also becomes related to the motif information. But, the related research on the directed weighted network needs to be further deepened. The weight of the network has a great influence on the structural attributes of the network, so it is necessary to extend the motif-based clustering to the weighted network. In this paper, a motif-based spectral clustering method for directed weighted networks is proposed. At the same time, this paper supplements the method of obtaining matrix expressions of the motif adjacency matrix in directed unweighted networks and provides a method to deal with the weight of networks, which will be helpful for the application research of motifs. This clustering method takes into account the higher-order connectivity patterns in networks and broadens the applicable range of spectral clustering to directed weighted networks. In this method, the motif-based clustering of directed weighted networks can be transformed into the clustering of the undirected weighted network corresponding to the motif-based adjacency matrix. The results show that the clustering method can correctly identify the partition structure of the benchmark network, and experiments on some real networks show that this method performs better than the method that does not consider the weight of networks. [ABSTRACT FROM AUTHOR]

Immediate post‐earthquake identification of structural damage is essential to prevent the loss of structural functionality and system failure. Vibration‐based damage identification methods have been widely implemented, but most tend to ignore a critical damage‐sensitive feature, the spatial correlation between sensor measurements. To this end, this paper proposes near‐real‐time damage identification by a graph convolutional autoencoder (GCAE) based on seismic responses of the structural system. The GCAE model accurately considers the spatial correlation and structural characteristics using a weighted adjacency matrix based on the structural modes. The proposed model consists of three main parts: (1) an "encoder" that learns latent features of the input data by considering the spatial information; (2) a "graph structure decoder" that detects anomalies in spatial correlation; and (3) a "node feature decoder" that captures changes in vibration signals. The GCAE model is trained to reconstruct structural responses of the target structure and the adjacency matrix in a healthy state. The seismic damage is then identified by the structural damage index calculated based on the difference between the input and reconstructed data. As numerical investigations, the proposed method is applied to two‐ and three‐dimensional steel frame structures. The train, validation, and test datasets are obtained by structural analyses using ground motions from the PEER‐NGA strong motion database. The proposed method is verified by the near‐real‐time simulation using the test dataset. The results show that the proposed GCAE model can accurately identify seismic damage in near‐real‐time. [ABSTRACT FROM AUTHOR]

Graph convolutional networks (GCNs), which extend convolutional neural networks (CNNs) to non-Euclidean structures, have been utilized to promote skeleton-based human action recognition research and have made substantial progress in doing so. However, there are still some challenges in the construction of recognition models based on GCNs. In this paper, we propose an enhanced adjacency matrix-based graph convolutional network with a combinatorial attention mechanism (CA-EAMGCN) for skeleton-based action recognition. Firstly, an enhanced adjacency matrix is constructed to expand the model's perceptive field of global node features. Secondly, a feature selection fusion module (FSFM) is designed to provide an optimal fusion ratio for multiple input features of the model. Finally, a combinatorial attention mechanism is devised. Specifically, our spatial-temporal (ST) attention module and limb attention module (LAM) are integrated into a multi-input branch and a mainstream network of the proposed model, respectively. Extensive experiments on three large-scale datasets, namely the NTU RGB+D 60, NTU RGB+D 120 and UAV-Human datasets, show that the proposed model takes into account both requirements of light weight and recognition accuracy. This demonstrates the effectiveness of our method. [ABSTRACT FROM AUTHOR]

Traffic prediction is important in applications such as traffic management, route planning, and traffic flow optimization. Traffic speed prediction is an important part of traffic forecasting, which has always been a challenging problem due to the complexity and dynamics of traffic systems. In order to predict traffic speed more accurately, we propose a traffic speed prediction model based on a multi-head attention mechanism and weighted adjacency matrix: MAT-WGCN. MAT-WGCN first uses GCN to extract the road spatial features in the weighted adjacency matrix, and it uses GRU to extract the correlation between speed and time from the original features. Then, the spatial features extracted by GCN and the temporal features extracted by GRU are fused, and a multi-head attention mechanism is introduced to integrate spatiotemporal features, collect and summarize spatiotemporal road information, and realize traffic speed prediction. In this study, the prediction performance of MAT-WGCN was tested on two real datasets, EXPY-TKY and METR-LA, and compared with the performance of traditional methods such as HA and SVR that do not combine spatial features, as well as T-GCN, A3T-GCN, and newer methods such as GCN and NA-DGRU that combine spatial features. The experimental results demonstrate that MAT-WGCN can capture the temporal and spatial characteristics of road conditions, thus enabling accurate traffic speed predictions. Furthermore, the incorporation of a multi-head attention mechanism significantly enhances the robustness of our model. [ABSTRACT FROM AUTHOR]

Let G be a connected graph. Denote by d i the degree of a vertex v i in G. Let f (x , y) > 0 be a real symmetric function. Consider an edge-weighted graph in such a way that for each edge v i v j of G, the weight of v i v j is equal to the value f (d i , d j) . Therefore, we have a degree-based weighted adjacency matrix A f (G) of G, in which the (i, j)-entry is equal to f (d i , d j) if v i v j is an edge of G and is equal to zero otherwise. Let x be a positive eigenvector corresponding to the largest eigenvalue λ 1 (A f (G)) of the weighted adjacency matrix A f (G) . In this paper, we first consider the unimodality of the eigenvector x on an induced path of G. Second, if f(x, y) is increasing in the variable x, then we investigate how the largest weighted adjacency eigenvalue λ 1 (A f (G)) changes when G is perturbed by vertex contraction or edge subdivision. The aim of this paper is to unify the study of spectral properties for the degree-based weighted adjacency matrices of graphs. [ABSTRACT FROM AUTHOR]

Entropy is widely applied to graph theory and complex networks as a powerful tool for measuring uncertainty in a complex system. Due to the fact that traditional probability distribution entropy cannot effectively characterize the global topology information of complex networks, some entropy measures constructed by the adjacency matrix A come into being, such as information-theoretic entropy EE and communicability sequence entropy. Despite substantial efforts to explore the properties of these measures, there remain some imperfections. For instance, the adjacency matrix only reflects the dependence between direct neighbors. Therefore, in this paper, we propose the r -power adjacency matrix entropy ( AME r ) to measure the indirect relationship between nodes in a network. And then, we compare the abilities of AME r , EE, and CSE in capturing the network global topology changes. Furthermore, we establish the Jenson–Shannon divergence based on AME r to quantify the structural dissimilarities of the networks. Finally, we apply the proposed methods to analyze the urban economic connection networks. The results demonstrate the availability of the proposed measures in identifying network topology changes and quantifying network structure differences. [ABSTRACT FROM AUTHOR]

Let A (G) be the adjacency matrix of a graph G. Let s (v) denote the row entries of A (G) corresponding to the vertex v of G. The Hamming distance between the strings s (u) and s (v) is the number of positions in which s (u) and s (v) differ. In this paper, we study the Hamming distance between the strings generated by the adjacency matrix of subgraph complement of a graph. We also compute sum of Hamming distances between all pairs of strings generated by the adjacency matrix of G ⊕ S. [ABSTRACT FROM AUTHOR]

Adjacency matrix visualization is a common method for presenting graph data, and the Focus+Context technique can be used to explore the details of the ROI (region of interest). Embedded views and multi-view approaches are usually applied when locating and comparing attributes among multiple nodes. However, the embedded view has an issue of edge occlusion, while the multi-view would cause repeated perspective switching. In this paper, we propose a Multivariate Fence (MVF) model as a focus view of the adjacency matrix to locate and compare attributes among nodes. An additional spatial parallel coordinate is added to the 2D adjacency matrix in an immersive environment so that the attribute information can be shown in a single view without blocking edge information. We also conduct a user study to evaluate the performance of the MVF. The results show that the MVF has better efficiency and accuracy in locating and comparing the multivariate adjacency matrix in the immersive environment against the existing focus model. Moreover, the MVF model is easier to understand and is preferred by users. [ABSTRACT FROM AUTHOR]

Based on the extended adjacency matrix, the spectral radius and the energy of the extended adjacency matrix are found that they possess high discriminating power and correlate well with a number of physicochemical properties and biological activities of organic compounds. In this research, by establishing a relationship between directed graphs and undirected graphs, we mainly present the Sachs theorem and the Coulson’s integral formula of the energy of the extended adjacency matrix of G. By using the formula, we can compare the energies of the extended adjacency matrices of two graphs and we find that the graph energy is less than the extended energy in the case of a tree. [ABSTRACT FROM AUTHOR]

Graph convolutional networks (GCNs) have become the de facto approaches and achieved state-of-the-art results for circumventing many real-world problems on graph-structured data. However, these networks are usually shallow due to the over-smoothing of GCNs with many layers, which limits the expressive power of learning graph representations. The current methods of solving the limitations have the bottlenecks of high complexity and many parameters. Although Simple Graph Convolution (SGC) reduces the complexity and parameters, it fails to distinguish the feature information of neighboring nodes at different distances. To tackle the limits, we propose MulStepNET, a stronger multi-step graph convolutional network architecture, that can capture more global information, by simultaneously combining multi-step neighborhoods information. When compared to existing methods such as GCN and MixHop, MulStepNET aggregates neighborhoods information at more distant distances via multi-power adjacency matrix while fitting fewest parameters and being computationally more efficient. Experiments on citation networks including Pubmed, Cora, and Citeseer demonstrate that the proposed MulStepNET model improves over SGC by 2.8, 3.3, and 2.1% respectively while keeping similar stability, and achieves better performance in terms of accuracy and stability compared to other baselines. [ABSTRACT FROM AUTHOR]

Accurate forecasting of taxi demand has facilitated the rational allocation of urban public transport resources, reduced congestion in urban transport networks, and shortened passenger waiting time. However, virtual station discovery and modelling of the demand when forecasting through graph convolutional neural networks remains challenging. In this study, the virtual station discovery problem was addressed by using a two-stage clustering approach, which considers the geographical and load characteristics of taxi demand. Furthermore, a fusion model combining non-negative matrix decomposition and a graph convolutional neural network was proposed in order to extract the features of the nodes for dimension reduction and adaptive adjacency matrix computation. By the construction of a local processing structure, further extraction of the local characteristics of the demand was achieved. The experimental results show that the method in this study outperforms state-of-the-art methods in terms of the root mean square error and average absolute value error. Therefore, the model proposed in this study is able to achieve accurate forecasting of taxi demand. [ABSTRACT FROM AUTHOR]

The research on node importance identification for temporal networks has attracted much attention. In this work, combined with the multi-layer coupled network analysis method, an optimized supra-adjacency matrix (OSAM) modeling method was proposed. In the process of constructing an optimized super adjacency matrix, the intra-layer relationship matrixes were improved by introducing the edge weight. The inter-layer relationship matrixes were formed by improved similarly and the inter-layer relationship is directional by using the characteristics of directed graphs. The model established by the OSAM method accurately expresses the structure of the temporal network and considers the influence of intra- and inter-layer relationships on the importance of nodes. In addition, an index was calculated by the average of the sum of the eigenvector centrality indices for a node in each layer and the node importance sorted list was obtained from this index to express the global importance of nodes in temporal networks. The experimental results on three real temporal network datasets Enron, Emaildept3, and Workspace showed that compared with the SAM and the SSAM methods, the OSAM method has a faster message propagation rate and larger message coverage and better SIR and NDCG@10 indicators. [ABSTRACT FROM AUTHOR]

In this paper, we introduced the adjacency matrix for flow problems and its cases, then we discussed how could we computed max flow by using this matrix. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n. The matrix S(G) = D(G) + A(G) is called the signless Laplacian matrix of G, where D(G) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let s1(G) and λ1(G) be the largest eigenvalue of S(G) and A(G), respectively. In this paper, we first present sharp upper and lower bounds for s1(G) and λ1(G) involving the maximum degree, the minimum degree, order, size and sum-connectivity F-index. Moreover, we investigate the relation between s1(G) and λ1(G). [ABSTRACT FROM AUTHOR]

Modeling complex spatial and temporal dependencies in multivariate time series data is crucial for traffic forecasting. Graph convolutional networks have proved to be effective in predicting multivariate time series. Although a predefined graph structure can help the model converge to good results quickly, it also limits the further improvement of the model due to its stationary state. In addition, current methods may not converge on some datasets due to the graph structure of these datasets being difficult to learn. Motivated by this, we propose a novel model named Dynamic Correlation Graph Convolutional Network (DCGCN) in this paper. The model can construct adjacency matrices from input data using a correlation coefficient; thus, dynamic correlation graph convolution is used for capturing spatial dependencies. Meanwhile, gated temporal convolution is used for modeling temporal dependencies. Finally, we performed extensive experiments to evaluate the performance of our proposed method against ten existing well-recognized baseline methods using two original and four public datasets. [ABSTRACT FROM AUTHOR]

Knowledge Graph (KG) reasoning has been an interesting topic in recent decades. Most current researches focus on predicting the missing facts for incomplete KG. Nevertheless, Temporal KG (TKG) reasoning, which is to forecast future facts, still faces with a dilemma due to the complex interactions between entities over time. This article proposes a novel intricate Spatiotemporal Dependency learning Network (STDN) based on Graph Convolutional Network (GCN) to capture the underlying correlations of an entity at different timestamps. Specifically, we first learn an adaptive adjacency matrix to depict the direct dependencies from the temporally adjacent facts of an entity, obtaining its previous context embedding. Then, a Spatiotemporal feature Encoding GCN (STE-GCN) is proposed to capture the latent spatiotemporal dependencies of the entity, getting the spatiotemporal embedding. Finally, a time gate unit is used to integrate the previous context embedding and the spatiotemporal embedding at the current timestamp to update the entity evolutional embedding for predicting future facts. STDN could generate the more expressive embeddings for capturing the intricate spatiotemporal dependencies in TKG. Extensive experiments on WIKI, ICEWS14, and ICEWS18 datasets prove our STDN has the advantage over state-of-the-art baselines for the temporal reasoning task. [ABSTRACT FROM AUTHOR]

Accurate urban flow prediction (UFP) is crucial for a range of smart city applications such as traffic management, urban planning, and risk assessment. To capture the intrinsic characteristics of urban flow, recent efforts have utilized spatial and temporal graph neural networks to deal with the complex dependence between the traffic in adjacent areas. However, existing graph neural network based approaches suffer from several critical drawbacks, including improper graph representation of urban traffic data, lack of semantic correlation modeling among graph nodes, and coarse-grained exploitation of external factors. To address these issues, we propose DiffUFP, a novel probabilistic graph-based framework for UFP. DiffUFP consists of two key designs: (1) a semantic region dynamic extraction method that effectively captures the underlying traffic network topology, and (2) a conditional denoising score-based adjacency matrix generator that takes spatial, temporal, and external factors into account when constructing the adjacency matrix rather than simply concatenation in existing studies. Extensive experiments conducted on real-world datasets demonstrate the superiority of DiffUFP over the state-of-the-art UFP models and the effect of the two specific modules. [ABSTRACT FROM AUTHOR]

In this paper, we define the adjacency matrix of a semigraph. We give the conditions for a matrix to be semigraphical and give an algorithm to construct a semigraph from the semigraphical matrices. We derive lower and upper bounds for largest eigenvalues. We study the eigenvalues of adjacency matrix of two types of star semigraphs. [ABSTRACT FROM AUTHOR]

Let G$$ G $$ be a finite tree. For any matching M$$ M $$ of G$$ G $$, let U(M)$$ U(M) $$ be the set of vertices uncovered by M$$ M $$. Let ℳG$$ {\mathcal{M}}_G $$ be a uniform random maximum size matching of G$$ G $$. In this paper, we analyze the structure of U(ℳG)$$ U\left({\mathcal{M}}_G\right) $$. We first show that U(ℳG)$$ U\left({\mathcal{M}}_G\right) $$ is a determinantal process. We also show that for most vertices of G$$ G $$, the process U(ℳG)$$ U\left({\mathcal{M}}_G\right) $$ in a small neighborhood of that vertex can be well approximated based on a somewhat larger neighborhood of the same vertex. Then we show that the normalized Shannon entropy of U(ℳG)$$ U\left({\mathcal{M}}_G\right) $$ can be also well approximated using the local structure of G$$ G $$. In other words, in the realm of trees, the normalized Shannon entropy of U(ℳG)$$ U\left({\mathcal{M}}_G\right) $$—that is, the normalized logarithm of the number of maximum size matchings of G$$ G $$—is a Benjamini‐Schramm continuous parameter. [ABSTRACT FROM AUTHOR]

In water distribution networks (WDNs), pipe failure is the most frequent type of equipment malfunction, resulting in water supply suspension. To determine the water suspension area and to properly plan restoration work, the accurate identification of isolated subsystems with respect to valve locations is critical. The water suspension area can be divided into intended and unintended isolations, depending on the valve layout. This technical note discusses the conventional method of searching the valve-controlled isolation areas and specifically examines the weakness of the unintended isolation (UI) search algorithm that uses a node-by-node adjacency matrix. In this study, we propose a new search algorithm using a link-by-link adjacency matrix to identify the UI segments. Demonstrations on branch and loop networks show that the new approach correctly identifies the UI in various operating conditions. The proposed method is expected to replace the conventional method and can be applied in various valve-controlled segment analyses, such as optimal valve placement, real-time valve control for dynamic WDN operation, and postaccident pipe recovery planning. [ABSTRACT FROM AUTHOR]

Exploring a general and effective method of isomorphism identification is an arduous task. For this aim, some new concepts such as binary link path and link-assortment adjacency matrix are introduced in this paper. On this basis, a new method is proposed to improve the operability of isomorphism identification and relieve the computational pressure. By generating new elements from structural features and connection relations among links or vertices in the graph, it transforms traditional high-ranking adjacency matrix into new low-ranking adjacency matrix. Some typical graphs including kinematic chains, topological graphs, and planetary gear trains are used to verify the effectiveness of the proposed method. As shown by a comparison with results in the cited references, the proposed method is available in isomorphism identification. [ABSTRACT FROM AUTHOR]

There is substantial interest in the current literature – spanning finance, economics, engineering and medical imaging – on the relationship structure between several nodes in a complex system. In this paper, we extend the literature by developing model and inference for complex networks in terms of latent factors, by extracting the hidden factors that plays significant role in the configuration of inter node relationships. Together, we extend inferences to applications where the underlying network structure is also latent; that is, the adjacency matrix is unobserved. Here, we consider a Bayesian variant of the matrix factorization technique to develop a structural model of the latent adjacency matrix. There are many potential applications. For illustration, we consider a latent network of firms in the in the US automotive sector, where the central object is to understand the impact of an economic shock on firms (or nodes of the network). An important question centers around the factors that affect the stability and resilience of inter node relationships in a network. In the automotive sector application, we would like to know whether these relationship structures are driven by collaborative or competitive environments? What are the effects of a collaborative or competitive role played by a specific firm on the configuration of the relationship network? Using accounting data on firm sales and costs, we use our proposed object oriented factorization methodology to provide explanation of the estimated network links between 3 major US auto manufacturers and their intermediate suppliers. [ABSTRACT FROM AUTHOR]

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]

In this short paper, we establish formulae to compute Wiener index, reciprocal Wiener index and peripheral Wiener index of graphs using adjacency matrix. Further, we present algorithms for the same. [ABSTRACT FROM AUTHOR]

The rank of adjacency matrix plays an important role in construction of linear codes from a directed strongly regular graph using different techniques, namely, code orthogonality, adjacency matrix determinant and adjacency matrix spectrum. The problem of computing the dimensions of such codes is an intriguing one. Several conjectures to determine the rank of adjacency matrix of a DSRG Γ over a finite field, keep researchers working in this area. To address the same to an extent, we have considered the problem of finding the rank over a finite field of the adjacency matrix of a DSRG Γ (v , k , t , λ , μ) with μ = 1 , including some mixed Moore graphs and corresponding codes arising from them, in this paper. [ABSTRACT FROM AUTHOR]

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of a complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron–Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of a gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same. [ABSTRACT FROM AUTHOR]

An extend adjacency matrix of a graph ( A e x ) was introduced decades ago as a precursor for developing a few quite useful molecular topological descriptors. The spectral radius ( η 1 ) of the extended adjacency matrix and the extended energy of a graph ( E e x ) have been successfully utilized in QSPR/QSAR investigations. Here, the η 1 and E e x have been further mathematically analyzed. Several sharp upper bounds for the E e x are obtained. In addition, the Nordhaus–Gaddum-type results for the η 1 and E e x are presented. [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]

The extended adjacency matrix Ae (G) of a graph G is determined by graph degrees. Some inequalities are found for the extended adjacency matrix including its vertices, its edges and its degrees in this paper. Also, some bounds are established for this matrix involving its energy and its laplacian energy. [ABSTRACT FROM AUTHOR]

Forecasting traffic flow is crucial for Intelligent Traffic Systems (ITS), traffic control, and traffic management systems. Complex spatial and temporal interactions of traffic networks make traffic forecasting tasks challenging. Recently, Graph Convolutional Network (GCN) has attracted researchers' attention as it can better represent graph-shaped road networks and extract spatial features of traffic. However, traditional GCN has some drawbacks since it uses a static adjacency matrix which is unable to capture the time-varying features of traffic propagation. To overcome this, we represent the traffic road network as a dynamic graph and use a probabilistic spatiotemporal adjacency matrix to identify the time-varying impacts of adjacent roads on target roads in GCN. In addition, to find the similarity among the nonadjacent nodes, we have employed node-specific learning in GCN rather than sharing parameters in traditional GCN. This node-specific learning helps our model to learn detailed characteristics of road networks. For temporal feature extractions, we used a Gated Recurrent Unit (GRU) that captures the local trend of traffic flow and an attention mechanism to capture the global trend of traffic flow. We compared the performance of our model with baseline models using two real-world datasets. Experimental results show that our model is effective in forecasting both short and long-term traffic flow. Source code of our model is available at https://github.com/atkia/PSTGCN [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]

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Topological indices are widely used for characterizing molecular graphs, establishing relationships between structure and properties of molecules. In this article we discuss adjacency matrix and three topological indices: Wiener index, Laplacian energy and Zagreb indices of zero-divisor graphs of Z n . We also provide a MATLAB code for Laplacian energy and Zagreb indices of Γ (Z n) . [ABSTRACT FROM AUTHOR]