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This paper presents a collaborative filtering algorithm based on reinforcement learning theory. Then, the personalized bank financial recommendation system for users is constructed in the massive data environment. Tags mimic different types of user interest points to build a representative personalized data set. The collaborative screening of bank financial products is realized using the simulation results and users' historical access records. The ranking calculation of related financial products is added to the general bank financial product recommendation system. This method can more accurately express the query results for a specific user. It is found that the collaborative filtering algorithm based on enhanced learning theory can improve the efficiency of collaborative screening of bank financial products. The best results can be obtained by combining the two organically. This paper proposes that the recommendation algorithm of reinforcement learning bank financial products based on user preference and collaborative filtering is feasible. [ABSTRACT FROM AUTHOR]

For the adjacency matrix A(G) and diagonal matrix of the vertex degrees D(G) of a simple graph G, the A(G) matrix is the convex combinations of D(G) and A(G), and is defined as A(G) = D(G)+(1)A(G), for 0 n be the eigenvalues of A(G) (which we call -adjacency eigenvalues of the graph G). The generalized adjacency energy also called -adjacency energy of the graph G is defined as EA (G) = is the average vertex degree, m is the size and n is the order of G. The -adjacency energy of a graph G merges the theory of energy (adjacency energy) and the signless Laplacian energy, as EA0 (G) = E (G) and 2E A 12 (G) = QE(G), where E (G) is the energy and QE(G) is the signless Laplacian energy of G. In this paper, we obtain some new upper and lower bounds for the generalized adjacency energy of a graph, in terms of different graph parameters like the vertex covering number, the Zagreb index, the number of edges, the number of vertices, etc. We characterize the extremal graphs attained 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]

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. [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]

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]

In this paper we construct several classes of non-regular graphs which are co-spectral with respect to all the three matrices, namely, adjacency, Laplacian and normalized Laplacian, and hence we answer a question asked by Butler [2]. We make these constructions starting with two pairs (G1, H1) and (G2, H2) of A-cospectral regular graphs, then considering the subdivision graphs S(Gi) and R-graphs R(Hi), i = 1, 2, and finally making some kind of partial joins between S(G1) and R(G2) and S(H1) and R(H2). Moreover, we determine the number of spanning trees and the Kirchhoff index of the newly constructed graphs. [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]

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]

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]

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]

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)

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]

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]

This paper established the idea of minimum evdominating matrix of semigraph and calculated its energy. The minimum ev-dominating energy EmeD(G) of a semigraph G is the sum of the absolute values of the eigenvalues of the minimum ev-dominating matrix. Here some results are also derived in connection with the energy of minimum evdominating matrix. Some lower bounds are also established. [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]

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 finite simple graph and let A(G) be its adjacency matrix. Then G is singular if A(G) is singular. Suppose Pb1, Pb2, Pb3 are three paths with disjoint vertices, where bi > 2(i = 1,2,3), and at most one of them is 2. Coalescing together one of the two end vertices of each of the three paths, and coalescing together the other end vertex of each of the three paths, the resulting graph is called the ϑ-graph, denoted by ϑ (b1, b2, b3). Let α(a, b1, b2, b3, s) be the graph obtained by merging one end of the path Ps with one vertex of a cycle Ca, and merging the other end of the path Ps with one vertex of ϑ(b1, b2, b3) of degree 3. If s = 1, denote β(a, b1, b2, b3) = α(a, b1, b2, b3, 1). In this paper, we give the necessity and sufficiency condition for the singularity of α(a, b1, b2, b3, s) and β(a, b1, b2, b3), and we also prove that the probability that any given α(a, b1, b2, b3, s) is a singular graph is equal to 35/64, the probability that any given β(a, b1, b2, b3) is a singular graph is equal to 9/16. From our main results we can conclude that such a &#945(a, b1, b2, b3, s) graph (&#946(a, b1, b2, b3) graph) is singular if 4|α or three bi(i = 1, 2, 3) are all odd numbers or exactly two of the three bi(i = 1, 2, 3) are odd numbers and the length of the cycle formed by the two odd paths in α(a, b1, b2, b3, s) graph (β(a, b1, b2, b3) graph) is a multiple of 4. The theoretical probability of these graphs being singular is more than half. [ABSTRACT FROM AUTHOR]

In this paper, we study the prime order Cayley graph assigned to the group ℤn for different values of n. We specify some of the graph theoretical properties such as chromatic and perfect matching numbers. Furthermore, we determine the adjacency matrices and eigenvalues of the prime order Cayley graph associated with groups ℤn and 풟2n. [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]

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]

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]

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.

Biclustering algorithm is an effective tool for processing gene expression datasets. There are two kinds of data matrices, binary data and non-binary data, which are processed by biclustering method. A binary matrix is usually converted from pre-processed gene expression data, which can effectively reduce the interference from noise and abnormal data, and is then processed using a biclustering algorithm. However, biclustering algorithms of dealing with binary data have a poor balance between running time and performance. In this paper, we propose a new biclustering algorithm called the Adjacency Difference Matrix Binary Biclustering algorithm (AMBB) for dealing with binary data to address the drawback. The AMBB algorithm constructs the adjacency matrix based on the adjacency difference values, and the submatrix obtained by continuously updating the adjacency difference matrix is called a bicluster. The adjacency matrix allows for clustering of gene that undergo similar reactions under different conditions into clusters, which is important for subsequent genes analysis. Meanwhile, experiments on synthetic and real datasets visually demonstrate that the AMBB algorithm has high practicability. [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]

Let S = (G, σ) be a signed graph, where σ is the sign function on the edges of the underlying graph G. It is widely recognized that the adjacency spectrum alone cannot uniquely determine a signed graph. Therefore, it is of great interest to identify whether there exist any cospectral, non-isomorphic signed graphs within a specific class of signed graphs. In a significant contribution, Godsil et al. demonstrated that two components of G1 × G2, where both G1 and G2 are connected bipartite graphs, are cospectral if and only if at the minimum one of G1 and G2 is balanced. In this paper, we first generalize Godsil’s result for two connected signed bipartite graphs S1 and S2. Furthermore, we will define partitioned tensor product of two signed bipartite graphs, which will enable us to generate multiple pairs of cospectral non-isomorphic signed bipartite graphs. [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]

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.

Connected planar graphs are of interest to a variety of scholars. Being able to simulate a database of such graphs with selected properties would support specific types of inference for spatial analysis and other network-based disciplines. This paper presents a simple, efficient, and flexible connected planar graph generator for this purpose. It also summarizes a comparison between an empirical set of specimen graphs and their simulated counterpart set, and establishes evidence for positing a conjecture about the principal eigenvalue of connected planar graphs. Finally, it summarizes a probability assessment of the algorithm’s outcomes as well as a comparison between the new algorithm and selected existing planar graph generators. [ABSTRACT FROM AUTHOR]

Let D(G) and A(G), respectively, be the diagonal matrix of vertex de- grees and the adjacency matrix of a connected graph G. The generalized adjacency matrix of G is defined as Aα(G) = αD(G) + (1 − α)A(G), α ∈ [0, 1]. The spread of the generalized adjacency matrix, denoted by S(Aα(G)), is defined as the difference of the largest and smallest eigenvalues of Aα(G). The spread S(Q(G)) of Q(G), the signless Laplacian matrix of G and S(A(G)) of A(G) are similarly defined. In this paper, we establish the relationships between S(Aα(G)), S(Q(G)) and S(A(G)). Further, we find bounds for S(Aα(G)) involving different parameters of G. Also, we obtain lower bounds for S(Aα(G)) in terms of the clique number and independence number of G, and we characterize the extremal graphs in some cases. [ABSTRACT FROM AUTHOR]

Graph colouring problem is a well-known NP-class optimization problem, studied due to a lot of applications in various real-world problems. Some of these applications are: register allocation, image processing and communication networks. There are various techniques suggested by the researchers to solve the problem which is either exact or approximate in nature. In this paper, a new greedy technique, based on degrees of vertices in the graph is presented to solve the graph colouring problem in an improved manner. The technique involves the use of adjacency matrix along with another matrix generated for the set of possible colours for each vertex in the graph. The generated colour matrix is used to assign the colours among the vertices in the graph based on decreasing degrees of the vertices. Several DIMACS colouring instances solved using the suggested approach in the article and compared with some contemporary techniques for the performance and proves compatible and having better execution time with compared technique. The obtained colouing results are mostly optimal colour values corresponding to the examined colouring instances of the graph. [ABSTRACT FROM AUTHOR]

How to mine many interesting subgraphs in uncertain graph has become an important research field in data mining. In this paper, a novel algorithm Uncertain Maximal Frequent Subgraph Mining Algorithm Based on Adjacency Matrix and Weight (UMFGAMW) is proposed. The definition of the adjacency matrix and the standard matrix coding for uncertain graph are presented. The correspondence between the adjacency matrix and uncertain graph is established. A new vertex ordering policy for computing the standard coding of uncertain graph adjacency matrix is designed. The complexity of uncertain graph standard coding is reduced, and the matching speed of uncertain subgraph standard coding is improved. The definition of the weight of uncertain graph and the mean weight of uncertain edge is proposed. The importance of the uncertain subgraphs that meet the minimum support threshold in the graph dataset is fully considered. Finally, a depth-first search weighted uncertain maximal frequent subgraph mining algorithm is discussed. According to the limiting condition of the uncertain maximum frequent subgraph and weighed uncertain edge, the number of mining results is reduced effectively. Experimental results demonstrate that the UMFGAMW algorithm has higher efficiency and better scalability. [ABSTRACT FROM AUTHOR]

The anti-Gallai graph Δ (G) of a graph G, has the edges of G as its vertices and two distinct edges are adjacent in Δ (G) if they lie on a triangle in G. In this paper we characterize the graphs whose anti-Gallai graph has only one or two eigenvalues and study the adjacency spectrum of the anti-Gallai graph of (K 4 , H) -free graphs, where H is the Hajos graph and graphs in which two diamonds share at most one vertex. We also find expressions for the adjacency spectrum of anti-Gallai graph of graphs after applying certain binary operations. [ABSTRACT FROM AUTHOR]

Gain graphs are graphs where the edges are given some orientation and labeled with the elements (called gains) from a group so that gains are inverted when we reverse the direction of the edges. Generalizing the notion of gain graphs, skew gain graphs have the property that the gain of a reversed edge is the image of edge gain under an anti-involution. In this paper, we study two different types, Laplacian and g-Laplacian matrices for a skew gain graph where the skew gains are taken from the multiplicative group Fx of a field F of characteristic zero. Defining incidence matrix, we also prove the matrix tree theorem for skew gain graphs in the case of the g-Laplacian matrix. [ABSTRACT FROM AUTHOR]

In an earlier paper, we have introduced the Tosha-degree of an edge in a graph without multiple edges and studied some properties. In this paper, we extend the definition of Tosha-degree of an edge in a graph in which multiple edges are allowed. Also, we introduce the concepts - zero edges in a graph, T-line graph of a multigraph, Tosha-adjacency matrix, Tosha-energy, edge- adjacency matrix and edge energy of a graph G and obtain some results. [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]

A Hamiltonian cycle in a graph is a cycle that visits each node/vertex exactly once. A graph containing a Hamiltonian cycle is called a Hamiltonian graph. There have been several researches to find the number of Hamiltonian cycles of a Hamilton graph. As the number of vertices and edges grow, it becomes very difficult to keep track of all the different ways through which the vertices are connected. Hence, analysis of large graphs can be efficiently done with the assistance of a computer system that interprets graphs as matrices. And, of course, a good and well written algorithm will expedite the analysis even faster. The most convenient way to quickly test whether there is an edge between two vertices is to represent graphs using adjacent matrices. In this paper, a new algorithm is proposed to find fuzzy Hamiltonian cycle using adjacency matrix and the degree of the vertices of a fuzzy graph. A fuzzy graph structure is also modeled to illustrate the proposed algorithms with the selected air network of Indigo airlines. [ABSTRACT FROM AUTHOR]

Graph coarsening is a widely used dimensionality reduction technique for approaching large-scale graph machine-learning problems. Given a large graph, graph coarsening aims to learn a smaller-tractable graph while preserving the properties of the originally given graph. Graph data consist of node features and graph matrix (e.g., adjacency and Laplacian). The existing graph coarsening methods ignore the node features and rely solely on a graph matrix to simplify graphs. In this paper, we introduce a novel optimization-based framework for graph dimensionality reduction. The proposed framework lies in the unification of graph learning and dimensionality reduction. It takes both the graph matrix and the node features as the input and learns the coarsen graph matrix and the coarsen feature matrix jointly while ensuring desired properties. The proposed optimization formulation is a multi-block non-convex optimization problem, which is solved efficiently by leveraging block majorization-minimization, log determinant, Dirichlet energy, and regularization frameworks. The proposed algorithms are provably convergent and practically amenable to numerous tasks. It is also established that the learned coarsened graph is ϵ 2 (0, 1) similar to the original graph. Extensive experiments elucidate the efficacy of the proposed framework for real-world applications. The code for all the experimental results is available at CODE. [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.

In this paper, we discuss various connections between the smallest eigenvalue of the adjacency matrix of a graph and its structure. There are several techniques for obtaining upper bounds on the smallest eigenvalue, and some of them are based on Rayleigh quotients, Cauchy interlacing using induced subgraphs, and Haemers interlacing with vertex partitions and quotient matrices. In this paper, we are interested in obtaining lower bounds for the smallest eigenvalue. Motivated by results on line graphs and generalized line graphs, we show how graph decompositions can be used to obtain such lower bounds. [ABSTRACT FROM AUTHOR]

This paper introduces a novel automatic modulation classification (AMC) method using the graph-based fractional lower-order cyclic-spectrum analysis in the $\alpha$ -stable noise environment. The noise in the practical communication scenario usually exhibits impulse characteristics in the statistical sense, which could be modeled as the $\alpha$ -stable distribution. This would make the second- or higher-order statistics of the received signal vanish, and thus the performances of the conventional AMC algorithms designed for Gaussian noise significantly deteriorate when directly employed in the $\alpha$ -stable noise. In our proposed framework, the fractional lower-order cyclic spectrum (FLOCS) analysis is first invoked to acquire the polyspectra of the signal corrupted by the $\alpha$ -stable noise. Then, the graph-based AMC mechanism is systematically established upon the graph representation of the FLOCS to identify the modulation type according to the discrepancies between the graph features derived from the training and test data. The performance of our proposed new algorithm is theoretically analyzed, and the correct classification probability $P_{\text{cc}}$ over the modulation candidate set is formulated analytically. The remarkable scalability and efficiency of our proposed approach for the modulation candidate set variation are also theoretically investigated in detail. Monte Carlo simulation results demonstrate the effectiveness and superiority of the proposed AMC scheme. [ABSTRACT FROM AUTHOR]

Let A(G) be the adjacency matrix of a graph G. Let s(vi) denote the row entries of A(G) corresponding to the vertex vi of G. The Hamming distance between the strings s(ui) and s(vi) is the number of positions in which their elements differ. The sum of Hamming distance between all the pairs of vertices is the Hamming index of a graph. In this paper, we study the Hamming distance between the strings generated by the adjacency matrix of various products of complete bipartite and complete graph. We also compute the Hamming index generated by the adjacency matrix of these graph products. [ABSTRACT FROM AUTHOR]

A mixed graph D is a graph that can be obtained from a graph by orienting some of its edges. Let α be a primitive nth root of unity, then the α−Hermitian adjacency matrix of a mixed graph is defined to be the matrix Hα = [hrs] where hrs = α if rs is an arc in D, hrs = α if sr is an arc in D, hrs = 1 if sr is a digon in D and hrs = 0 otherwise. In this paper we study the cospectrality of the Hermitian adjacency matrix of a mixed graph. [ABSTRACT FROM AUTHOR]

In this paper, a serial reconfigurable modular robot is presented. Each module is a cubic structure. Each surface of this cube has a connection interface. It has three rotational degrees of freedom, which are distributed along the three directions of its local coordinate system. A new and reliable docking mechanism is designed to realize the connection among adjacent modules. With the novel structure, the vibration of the modular robot is negligible. Then, based on the characteristics of the module, an eigenvector to describe the state information of a single module and an adjacency matrix to express the topology structure of a multi-module system are defined. Based on the screw theory, a kinematic model of the serial modular robot is proposed. Using an exponential product formula, the modular expressions of the kinematic equations are obtained automatically. According to the motion characteristics of each module and the structure characteristics of the modular robot, we can simplify the module which provides redundant degrees of freedom to a connecting rod. This method can reduce the complexity of solving the inverse kinematic solution. To solve the inverse kinematics automatically, we decompose it into two sub-motion models and propose the conditions of applying the subproblem method. Finally, a simulation by MATLAB is used to verify the correctness of the automatic generation of the forward and inverse kinematics. [ABSTRACT FROM AUTHOR]