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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]

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]

Aspect-level sentiment analysis is used to predict the sentiment polarity of a specific aspect in a sentence. However, most current research cannot fully utilize semantic information, and the models lack robustness. Therefore, this article proposes a model for aspect-level sentiment analysis based on a combination of adversarial training and dependency syntax analysis. First, BERT is used to transform word vectors and construct adjacency matrices with dependency syntactic relationships to better extract semantic dependency relationships and features between sentence components. A multi-head attention mechanism is used to fuse the features of the two parts, simultaneously perform adversarial training on the BERT embedding layer to enhance model robustness, and, finally, to predict emotional polarity. The model was tested on the SemEval 2014 Task 4 dataset. The experimental results showed that, compared with the baseline model, the model achieved significant performance improvement after incorporating adversarial training and dependency syntax relationships. [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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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]

The Rubik's Cube mechanism (RCM) is a kind of reconfigurable mechanism with multiple characteristics such as multiple configurations, variable topology, strong coupling, and reconfigurability. Crossover research on the RCM with mathematics, chemistry, cryptography, and other disciplines has led to important breakthroughs and progress. It is obvious that the invention and creation of a new RCM can provide important ideological inspiration and theoretical guidance for the accelerated iterative updating of Rubik's Cube products and the expansion of their applications. This paper investigates the type synthesis method for a regular hexagonal prism (RHP) RCM (RHPRCM). Through analysis of the reconfigurable movement process of the RCM, two mechanism factors are abstracted, a type synthesis process for the RHPRCM is proposed, a symmetry layout method for the RCM's revolute axis based on the RHP space polyhedron is proposed, and an analysis method for the intersection of the revolute pair contact surfaces (RPCSs) based on the adjacency matrix is proposed. Taking a revolute axis passing through the center of an RHP and having only one RPCS for each revolute axis as an example, an RHPRCM with different topological structures is synthesized. The relevant research in this paper can provide methodological guidance for the synthesis of other spatial RCMs. [ABSTRACT FROM AUTHOR]

Two linear algebra problems implore a solution to them, creating the themes pursued in this paper. The first problem interfaces with graph theory via binary 0-1 adjacency matrices and their Laplacian counterparts. More contemporary spatial statistics/econometrics applications motivate the second problem, which embodies approximating the eigenvalues of massively large versions of these two aforementioned matrices. The proposed solutions outlined in this paper essentially are a reformulated multiple linear regression analysis for the first problem and a matrix inertia refinement adapted to existing work for the second problem. [ABSTRACT FROM AUTHOR]

Given that fingerprint localization methods can be effectively modeled as supervised learning problems, machine learning has been employed for indoor localization tasks based on fingerprint methods. However, it is often challenging for popular machine learning models to effectively capture the unstructured data features inherent in fingerprint data that are generated in diverse propagation environments. In this paper, we propose an indoor localization algorithm based on a high-order graph neural network (HoGNNLoc) to enhance the accuracy of indoor localization and improve localization stability in dynamic environments. The algorithm first designs an adjacency matrix based on the spatial relative locations of access points (APs) to obtain a graph structure; on this basis, a high-order graph neural network is constructed to extract and aggregate the features; finally, the designed fully connected network is used to achieve the regression prediction of the location of the target to be located. The experimental results on our self-built dataset show that the proposed algorithm achieves localization accuracy within 1.29 m at 80% of the cumulative distribution function (CDF) points. The improvements are 59.2%, 51.3%, 36.1%, and 22.7% compared to the K-nearest neighbors (KNN), deep neural network (DNN), simple graph convolutional network (SGC), and graph attention network (GAT). Moreover, even with a 30% reduction in fingerprint data, the proposed algorithm exhibits stable localization performance. On a public dataset, our proposed localization algorithm can also show better performance. [ABSTRACT FROM AUTHOR]

To meet the increased customer demands, microgrid evolved. The structure of microgrid changes dynamically due to the intermittent nature of renewable-based generation, status of the distributed generator and opening of breakers for fault/maintenance. Hence, the magnitude of fault current is dynamic in nature. In order to deal with these dynamic changes, this paper addresses an adaptive central microgrid controller-based protection and relay coordination scheme, which revises the relay settings dynamically (both radial and looped configuration) for every change in topology. In the proposed algorithm, the primary relay responds to a fault immediately since the individual relays are given with fault level setting. For any abnormality in the network, the fault location is determined both via local relay and microgrid central controller (MCC). Hence, even though the local relay fails to identify the fault due to high fault impedance, the MCC locates the fault accurately and isolates the minimum faulty part. The coordination between relays is carried out by MCC in a time-graded manner based on microgrid central protection and relay coordination algorithm. The proposed algorithm is tested using Matlab in a microgrid built based on the IEEE 33 bus distribution network. [ABSTRACT FROM AUTHOR]

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

Applying machine learning algorithms to graph-structured data has garnered significant attention in recent years due to the prevalence of inherent graph structures in real-life datasets. However, the direct application of traditional deep learning algorithms, such as Convolutional Neural Networks (CNNs), is limited as they are designed for regular Euclidean data like 2D grids and 1D sequences. In contrast, graph-structured data are in a non-Euclidean form. Graph Neural Networks (GNNs) are specifically designed to handle non-Euclidean data and make predictions based on connectivity rather than spatial structure. Real-life graph data can be broadly categorized into two types: spatially-invariant graphs, where the link structure between nodes is independent of their spatial positions, and spatially-variant graphs, where node positions provide additional information about the graph's properties. However, there is limited understanding of the effect of spatial variance on the performance of Graph Neural Networks. In this study, we aim to address this issue by comparing the performance of GNNs and CNNs on spatially-variant and spatially-invariant graph data. In the case of spatially-variant graphs, when represented as adjacency matrices, they can exhibit Euclidean-like spatial structure. Based on this distinction, we hypothesize that CNNs may outperform GNNs when working with spatially-variant graphs, while GNNs may excel on spatially-invariant graphs. To test this hypothesis, we compared the performance of CNNs and GNNs under two scenarios: (i) graphs in the training and test sets had the same connectivity pattern and spatial structure, and (ii) graphs in the training and test sets had the same connectivity pattern but different spatial structures. Our results confirmed that the presence of spatial structure in a graph allows for the effective use of CNNs, which may even outperform GNNs. Thus, our study contributes to the understanding of the effect of spatial graph structure on the performance of machine learning methods and allows for the selection of an appropriate algorithm based on the spatial properties of the real-life graph dataset. [ABSTRACT FROM AUTHOR]

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

Let G be a graph with the adjacency matrix A (G) , and let D (G) be the diagonal matrix of the degrees of G. Nikiforov first defined the matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G) , 0 ≤ α ≤ 1 , which shed new light on A (G) and Q (G) = D (G) + A (G) , and yielded some surprises. The α − adjacency energy E A α (G) of G is a new invariant that is calculated from the eigenvalues of A α (G) . In this work, by combining matrix theory and the graph structure properties, we provide some upper and lower bounds for E A α (G) in terms of graph parameters (the order n, the edge size m, etc.) and characterize the corresponding extremal graphs. In addition, we obtain some relations between E A α (G) and other energies such as the energy E (G) . Some results can be applied to appropriately estimate the α -adjacency energy using some given graph parameters rather than by performing some tedious calculations. [ABSTRACT FROM AUTHOR]

In road networks, attribute information carried by road segment nodes, such as weather and points of interest (POI), exhibit strong heterogeneity and often involve one-to-many or many-to-one relationships. However, research on such heterogeneity in traffic prediction is relatively limited. Our research examines how varying the network propagation pattern based on the degree of node-to-node heterogeneity of information affects the model prediction performance. Specifically, at the node level, we use knowledge embedding to generate knowledge vectors that quantify the heterogeneity among the attribute information of a node. At the road network level, we calculate a homogeneity adjacency matrix that captures both the topological structure of the road network and the similarity of node heterogeneity. This adjacency matrix assigns different weights to neighbors based on their homogeneity, guiding the propagation of graph convolutional networks (GCN). Finally, we separate the representation of propagation into self-representation and neighbor representation to extract multi-attribute information, including self, homogeneity, and heterogeneity. Experiments on real datasets demonstrate that the incorporation of our homogeneity adjacency matrix leads to a significant improvement in the accuracy of short-term and long-term prediction compared with previous work on homogeneous and single-dimensional information. Furthermore, our approach maintains its performance advantage over baseline models under different embedding dimensions and parameter settings. [ABSTRACT FROM AUTHOR]

For a simple graph with vertex set { v 1 , v 2 , ... , v n } with degree sequence d v i of vertex v i , i = 1 , 2 , ... , n , the inverse sum indeg matrix ( I S I -matrix) A I S I (G) = (a i j) n × n of G is defined by a i j = d v i d v j d v i + d v j , if v i is adjacent to v j , and zero, otherwise. The multiset of eigenvalues of A I S I (G) is the I S I -spectrum of G and the sum of their absolute values is the I S I -energy of G . In this paper, we modify the two results of (Li, Ye and Broersma, 2022), give the correct characterization of the extremal graphs and thereby obtain better bounds than the already known results. Moreover, we also discuss the QSPR analysis and carry the statistical modelling (linear, logarithmic and quadratic) of the physicochemical properties of anticancer drugs with the I S I -index (energy). [ABSTRACT FROM AUTHOR]

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

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]

Let G be a simple graph with the vertex set V = { v 1 , ... , v n } and denote by d v i the degree of the vertex v i . The modified Sombor index of G is the addition of the numbers (d v i 2 + d v j 2) − 1 / 2 over all of the edges v i v j of G. The modified Sombor matrix A M S (G) of G is the n by n matrix such that its (i , j) -entry is equal to (d v i 2 + d v j 2) − 1 / 2 when v i and v j are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of A M S (G) . The sum of the absolute eigenvalues of A M S (G) is known as the modified Sombor energy of G. Two graphs with the same modified Sombor energy are referred to as modified Sombor equienergetic graphs. In this article, several bounds for the modified Sombor index, the modified Sombor spectral radius, and the modified Sombor energy are found, and the corresponding extremal graphs are characterized. By using computer programs (Mathematica and AutographiX), it is found that there exists only one pair of the modified Sombor equienergetic chemical graphs of an order of at most seven. It is proven that the modified Sombor energy of every regular, complete multipartite graph is 2 ; this result gives a large class of the modified Sombor equienergetic graphs. The (linear, logarithmic, and quadratic) regression analyses of the modified Sombor index and the modified Sombor energy together with their classical versions are also performed for the boiling points of the chemical graphs of an order of at most seven. [ABSTRACT FROM AUTHOR]

We presented a method based on multigraphs to mathematically define a distribution function in time for the generation of data exchange in a special-purpose communication network. This is needed for the modeling and design of communication networks (CNs) consisting of integrated telecommunications and computer networks (ITCN). Simulation models require a precise definition of network traffic communication. An additional problem for describing the network traffic in simulation models is the mathematical model of data distribution, according to which the generation and exchange of certain types and quantities of data are realized. The application of multigraphs enabled the time and quantity of the data distribution to be displayed as operational procedures for a special-purpose communication unit. A multigraph was formed for each data-exchange time and allowed its associated adjacency matrix to be defined. Using the matrix estimation method allowed the mathematical definition of the distribution function values. The application of the described method for the use of multigraphs enabled a more accurate mathematical description of real traffic in communication networks. [ABSTRACT FROM AUTHOR]

Electroencephalography (EEG) is recorded by electrodes from different areas of the brain and is commonly used to measure neuronal activity. EEG-based methods have been widely used for emotion recognition recently. However, most current methods for EEG-based emotion recognition do not fully exploit the relationship of EEG channels, which affects the precision of emotion recognition. To address the issue, in this paper, we propose a novel method for EEG-based emotion recognition called CR-GCN: Channel-Relationships-based Graph Convolutional Network. Specifically, topological structure of EEG channels is distance-based and tends to capture local relationships, and brain functional connectivity tends to capture global relationships among EEG channels. Therefore, in this paper, we construct EEG channel relationships using an adjacency matrix in graph convolutional network where the adjacency matrix captures both local and global relationships among different EEG channels. Extensive experiments demonstrate that CR-GCN method significantly outperforms the state-of-the-art methods. In subject-dependent experiments, the average classification accuracies of 94.69% and 93.95% are achieved for valence and arousal. In subject-independent experiments, the average classification accuracies of 94.78% and 93.46% are obtained for valence and arousal. [ABSTRACT FROM AUTHOR]

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

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

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

The internet's rapid growth has resulted in an increase in the number of malicious files. Recently, powershell scripts and Windows portable executable (PE) files have been used in malicious behaviors. To solve these problems, artificial intelligence (AI) based malware detection methods have been widely studied. Among AI techniques, the graph convolution network (GCN) was recently introduced. Here, we propose a malicious powershell detection method using a GCN. To use the GCN, we needed an adjacency matrix. Therefore, we proposed an adjacency matrix generation method using the Jaccard similarity. In addition, we show that the malicious powershell detection rate is increased by approximately 8.2% using GCN. [ABSTRACT FROM AUTHOR]

This paper presents a novel theory and method to calculate the canonical labelings of digraphs whose definition is entirely different from the traditional definition of Nauty. It indicates the mutual relationships that exist between the canonical labeling of a digraph and the canonical labeling of its complement graph. It systematically examines the link between computing the canonical labeling of a digraph and the k-neighborhood and k-mix-neighborhood subdigraphs. To facilitate the presentation, it introduces several concepts including mix diffusion outdegree sequence and entire mix diffusion outdegree sequences. For each node in a digraph G, it assigns an attribute m_NearestNode to enhance the accuracy of calculating canonical labeling. The four theorems proved here demonstrate how to determine the first nodes added into MaxQ(G). Further, the other two theorems stated below deal with identifying the second nodes added into MaxQ(G). When computing Cmax(G), if MaxQ(G) already contains the first i vertices u1, u2, ..., ui, Diffusion Theorem provides a guideline on how to choose the subsequent node of MaxQ(G). Besides, the Mix Diffusion Theorem shows that the selection of the (i + 1)th vertex of MaxQ(G) for computing Cmax(G) is from the open mix-neighborhood subdigraph N++(Q) of the nodes set Q = {u1, u2, ..., ui}. It also offers two theorems to calculate the Cmax(G) of the disconnected digraphs. The four algorithms implemented in it illustrate how to calculate MaxQ(G) of a digraph. Through software testing, the correctness of our algorithms is preliminarily verified. Our method can be utilized to mine the frequent subdigraph. We also guess that if there exists a vertex v ∈ S+(G) satisfying conditions Cmax(G - v) ≤ Cmax(G - w) for each w ∈ S+(G) ^ w ≠ v, then u1 = v for MaxQ(G). [ABSTRACT FROM AUTHOR]

In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. Let G1,nc and G2,ncbe the classes of the connected graphs of order n whose complements are bicyclic with exactly two and three cycles, respectively. In this paper, we characterize the unique minimizing graph among all the graphs which belong to Gnc= G1,nc [ G2,nc, a class of the connected graphs of order n whose complements are bicyclic. [ABSTRACT FROM AUTHOR]

In this paper, we suggest an approach to recovering behavior-related, preference-choice network information from observational data. We model the process as a self-organized behavior based random exponential network-graph system. To address the unknown nature of the sampling model in recovering behavior related network information, we use the Cressie-Read (CR) family of divergence measures and the corresponding information theoretic entropy basis, for estimation, inference, model evaluation, and prediction. Examples are included to clarify how entropy based information theoretic methods are directly applicable to recovering the behavioral network probabilities in this fundamentally underdetermined ill posed inverse recovery problem. [ABSTRACT FROM AUTHOR]

In this paper, we present an innovative transmit power control scheme, based on optimization theory, for wireless sensor networks (WSNs) which use carrier sense multiple access (CSMA) with collision avoidance (CA) as medium access control (MAC) protocol. In particular, we focus on schemes where several remote nodes send data directly to a common access point (AP). Under the assumption of finite overall network transmit power and low traffic load, we derive the optimal transmit power allocation strategy that minimizes the packet error rate (PER) at the AP. This approach is based on modeling the CSMA/CA MAC protocol through a finite state machine and takes into account the network adjacency matrix, depending on the transmit power distribution and determining the network connectivity. It will be then shown that the transmit power allocation problem reduces to a convex constrained minimization problem. Our results show that, under the assumption of low traffic load, the power allocation strategy, which guarantees minimal delay, requires the maximization of network connectivity, which can be equivalently interpreted as the maximization of the number of non-zero entries of the adjacency matrix. The obtained theoretical results are confirmed by simulations for unslotted Zigbee WSNs. [ABSTRACT FROM AUTHOR]

The Estrada index of a graph G is defined as E E (G) = ∑ i = 1 n e λ i , where λ 1 , λ 2 , ... , λ n are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let U (n , d) be the set of all unicyclic graphs with n vertices and diameter d. In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among U (n , d) . We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even. [ABSTRACT FROM AUTHOR]

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

For α ∈ [ 0 , 1 ] , let A α (G σ) = α D (G) + (1 − α) A (G σ) , where G is a simple undirected graph, D (G) is the diagonal matrix of its vertex degrees and A (G σ) is the adjacency matrix of the signed graph G σ whose underlying graph is G. In this paper, basic properties of A α (G σ) are obtained, its positive semidefiniteness is studied and some bounds on its eigenvalues are derived—in particular, lower and upper bounds on its largest eigenvalue are obtained. [ABSTRACT FROM AUTHOR]

The rapid development of ubiquitous mobile computing has enabled the collection of new types of massive traffic data to understand collective movement patterns in social spaces. Contributing to the understanding of crowd formation and dispersal in populated areas, we developed a model of visitors' dynamic agglomeration patterns at a particular event using dynamic population data. This information, a type of big data, comprised aggregate Global Positioning System (GPS) location data automatically collected from mobile phones without users' intervention over a grid with a spatial resolution of 250 m. Herein, spatial autoregressive models with two-step adjacency matrices are proposed to represent visitors' movement between grids around the event site. We confirmed that the proposed models had a higher goodness-of-fit than those without spatial or temporal autocorrelations. The results also show a significant reduction in accuracy when applied to prediction with estimated values of the endogenous variables of prior time periods. [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of A (G) and D (G) and defined the A α -matrix for every real α ∈ [ 0 , 1 ] as: A α (G) = α D (G) + (1 − α) A (G). In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the A α -matrix with the Estrada index of the adjacency matrix of the graph G. Furthermore, lower bounds for the Estrada index are established. [ABSTRACT FROM AUTHOR]

Let A (G) be the adjacent matrix and D (G) the diagonal matrix of the degrees of a graph G, respectively. For 0 ≤ α ≤ 1 , the A α -matrix is the general adjacency and signless Laplacian spectral matrix having the form of A α (G) = α D (G) + (1 − α) A (G) . Clearly, A 0 (G) is the adjacent matrix and 2 A 1 2 is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The A α -spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results. [ABSTRACT FROM AUTHOR]

The maximum eigenvalue of the adjacency matrix (AM) has been supposed to contain rich information about the corresponding network. An experimental study focused on revealing the meaning and application of the maximum eigenvalue is missing. To this end, AM was constructed using mutual information (MI) to determine the functional connectivity with electroencephalogram (EEG) data recorded with a mental fatigue model, and then was converted into both binary and weighted brain functional network (BFN) and corresponding random networks (RNs). Both maximum eigenvalue and corresponding network characters in BFNs and RNs were considered to explore the changes during the formation of mental fatigue. The results indicated that large maximum eigenvalue means more edges in the corresponding network, along with a high degree and a short characteristic path length both in weighted and binary BFNs. Interestingly, the maximum eigenvalue of AM was always a little larger than that of the corresponding random matrix (RM), and had an obvious linearity with the sum of the AM elements, indicating that the maximum eigenvalue can be able to distinguish the network structures which have the same mean degree. What is more, the maximum eigenvalue, which increased with the deepening of mental fatigue, can become a good indicator for mental fatigue estimation. [ABSTRACT FROM AUTHOR]

This paper puts forward an innovative theory and method to calculate the canonical labelings of graphs that are distinct to N a u t y 's. It shows the correlation between the canonical labeling of a graph and the canonical labeling of its complement graph. It regularly examines the link between computing the canonical labeling of a graph and the canonical labeling of its o p e n k- n e i g h b o r h o o d s u b g r a p h . It defines d i f f u s i o n d e g r e e s e q u e n c e s and e n t i r e d i f f u s i o n d e g r e e s e q u e n c e . For each node of a graph G, it designs a characteristic m _ N e a r e s t N o d e to improve the precision for calculating canonical labeling. Two theorems established here display how to compute the first nodes of M a x Q (G) . Another theorem presents how to determine the second nodes of M a x Q (G) . When computing C m a x (G) , if M a x Q (G) already holds the first i nodes u 1 , u 2 , ⋯ , u i , Diffusion and Nearest Node theorems provide skill on how to pick the succeeding node of M a x Q (G) . Further, it also establishes two theorems to determine the C m a x (G) of disconnected graphs. Four algorithms implemented here demonstrate how to compute M a x Q (G) of a graph. From the results of the software experiment, the accuracy of our algorithms is preliminarily confirmed. Our method can be employed to mine the frequent subgraph. We also conjecture that if there is a node v ∈ S (G) meeting conditions C m a x (G − v) ⩽ C m a x (G − w) for each w ∈ S (G) ∧ w ≠ v , then u 1 = v for M a x Q (G) . [ABSTRACT FROM AUTHOR]

Online social networks have provided a promising communication platform for an activity inherently dear to the human heart, to find friends. People are recommended to each other as potential future friends by comparing their profiles which require numerical quantifiers to determine the extent of user similarity. From similarity-based methods to artificial intelligent machine learning methods, several metrics enable us to characterize social networks from different perspectives. This research focuses on the collaborative employment of neighbor based and graphical distance-based similarity measurement methods with text classification tools such as the feature matrix and feature vector. Likeminded nodes are predicted accurately and effectively as compared to other methods. [ABSTRACT FROM AUTHOR]

In this paper, multiple resolvable group target tracking was considered in the frame of random finite sets. In particular, a group target model was introduced by combining graph theory with the labeled random finite sets (RFS). This accounted for dependence between group members. Simulations were presented to verify the proposed algorithm. [ABSTRACT FROM AUTHOR]

Energies of molecular graphs have various applications in chemistry, polymerization, computer networking and pharmacy. In this paper, we give general closed forms of distance and adjacency energies of generalized wheel networks W n , m . Consequently, we give these results for classical wheel graphs. We also give pictorial dependencies of energies on the involved parameters m ≥ 3 and n . [ABSTRACT FROM AUTHOR]

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