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Let G be a simple graph with the adjacency matrix $ A(G) $ A (G). Let $ \tau (G) $ τ (G) denote the smallest positive eigenvalue of $ A(G) $ A (G). In 1990, Pavlíková and Kr $ \breve{c} $ c ˘ -Jediný proved that among all nonsingular trees on n = 2m vertices, the comb graph (obtained by taking a path on m vertices and adding a new pendant vertex to every vertex of the path) has the maximum τ value. We consider the problem for unicyclic graphs. Let $ \mathscr {U} $ U denote the class of all connected bipartite unicyclic graphs with a unique perfect matching, and for each $ m\geq ~3 $ m ≥ 3 , let $ \mathscr {U}_n $ U n be the subclass of $ \mathscr {U} $ U with graphs on n = 2m vertices. We first obtain the classes of unicyclic graphs U in $ \mathscr {U} $ U such that $ \tau (U)\leq \sqrt {2}-1 $ τ (U) ≤ 2 − 1. We then find the unique graph $ U_o^n $ U o n (resp. $ U_e^n $ U e n ) having the maximum τ value among all graphs in $ \mathscr {U}_n $ U n when m is odd (resp. when m is even). Finally, we prove that $ U_o^6 $ U o 6 (the graph obtained from a cycle of order 4, by adding two pendants to two adjacent vertices) is the graph with maximum τ value among all graphs in $ \mathscr {U} $ U . As a consequence, we obtain a sharp upper bound for $ \tau (U) $ τ (U) when $ U\in \mathscr {U} $ U ∈ U . [ABSTRACT FROM AUTHOR]

The (k , r) -fan, denoted by F k , r , is the graph consisting of k copies of the complete graph K r which intersect in a single vertex. Desai et al. [7] proved that E X s p (n , F k , r) ⊆ E X (n , F k , r) for sufficiently large n , where E X s p (n , F k , r) and E X (n , F k , r) are the sets of n -vertex F k , r -free graphs with maximum spectral radius and maximum size, respectively. In this paper, the set E X s p (n , F k , r) is uniquely determined for n large enough. Let H s , t 1 , ... , t k be the graph consisting of s triangles and k odd cycles of lengths t 1 , ... , t k ≥ 5 intersecting in exactly one common vertex, denoted by H s , k for short. Li and Peng [12] showed that E X s p (n , H s , k) ⊆ E X (n , H s , k) for n large enough. In this paper, the set E X s p (n , H s , k) is uniquely characterized for sufficiently large n. [ABSTRACT FROM AUTHOR]

Let G be a finite simple graph with λ as an eigenvalue (i.e. an eigenvalue of the adjacency matrix of G), and let H be a star complement for λ in G. Motivated by a controllability condition, we say that H is a strong star complement for λ if G and H have no eigenvalue in common. We explore this concept in the context of line graphs, exceptional graphs, strongly regular graphs and graphs with a prescribed star complement. [ABSTRACT FROM AUTHOR]

The comaximal graph Γ (R) of a commutative ring R is a simple graph with vertex set R and two distinct vertices u and v of Γ (R) are adjacent if and only if u R + v R = R. In this paper, we find the sharp bounds for the Sombor index for comaximal graphs of integer modulo ring ℤ n and give the corresponding extremal graphs. Also, we find the Sombor eigenvalues and the bounds for the Sombor energy of comaximal graphs of ℤ n . [ABSTRACT FROM AUTHOR]

Let G be a simple graph with the adjacency matrix A (G) , and let τ (G) denote the smallest positive eigenvalue of A (G). Let G n be the class of all connected bipartite graphs on n = 2 k vertices with a unique perfect matching. In this article, we characterize the graphs G in G n such that τ (G) does not exceed 1 2. Using the above characterization, we obtain the unique graphs in G n with the maximum and the second maximum τ , respectively. Further, we prove that the largest and the second largest limit points of the smallest positive eigenvalues of bipartite graphs with a unique perfect matching are 1 2 and the reciprocal of α 3 1 2 + α 3 − 1 2 , respectively, where α 3 is the largest root of x 3 − x − 1. [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]

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

A graph is called unimodular if its adjacency matrix has determinant ± 1. This article provides a necessary and sufficient condition for a simple connected graph with a unique perfect matching to be unimodular. In particular, we give a complete characterization of bicyclic unimodular graphs with a unique perfect matching. Moreover, the possible values of the determinant of the adjacency matrix of unicyclic, bicyclic, and tricyclic graphs with a unique perfect matching are also provided in this article. For non-bipartite unicyclic graphs with a unique perfect matching, we address the problem of when the inverse of the corresponding adjacency matrix is diagonally similar to a non-negative matrix. A pseudo-unimodular graph is a singular graph whose product of non-zero eigenvalues of the corresponding adjacency matrix is ± 1. We supply a necessary and sufficient condition for a singular graph to be pseudo-unimodular. [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]

Let (G, w) be a weighted graph with the adjacency matrix A. The group inverse of (G, w), denoted by (G # , w #) is the weighted graph with the weight w # (v i v j) of an edge v i v j in G # is defined as the ijth entry of A # , the group inverse of A. We study the group inverse of singular weighted trees. It is shown that if (T, w) is a singular weighted tree, then (T # , w #) is again a weighted tree if and only if (T, w) is a star tree, which in turn holds if and only if (T # , w #) is graph isomorphic to (T, w). A new class T w of weighted trees is introduced and studied here. It is shown that the group inverse of the adjacency matrix of a positively weighted tree in T w is signature similar to a non-negative matrix. [ABSTRACT FROM AUTHOR]

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

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]

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

In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra B n of order 2 n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ (P) of complement-closed and convex subposets P of B n. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V (P) on P. We then write down the adjacency operator A of the graph Γ (P) as the composition of two linear operators on V (P) , namely, the operator that multiplies elements of V (P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P , then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ (P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property. [ABSTRACT FROM AUTHOR]

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

We give two methods for a construction of self-orthogonal linear codes from equitable partitions of distance-regular graphs. By applying these methods, we construct self-orthogonal codes from equitable partitions of the graph of unitals in $ PG(2,4) $ and the only known strongly regular graph with parameters $ (216,40,4,8) $. Some of the codes obtained are optimal. [ABSTRACT FROM AUTHOR]

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

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

This study aims to compare the longitudinal performance between machine learning and deep learning classifiers for Android malware detection, employing different levels of feature abstraction. Using a dataset of 200k Android apps labeled by date within a 10-year range (2013-2022), we propose the LongCGDroid, an image-based effective approach for Android malware detection. We use the semantic Call Graph API representation that is derived from the Control Flow Graph and Data Flow Graph to extract abstracted API calls. Thus, we evaluate the longitudinal performance of LongCGDroid against API changes. Different models are used, machine learning models (LR, RF, KNN, SVM) and deep learning models (CNN, RNN). Empirical experiments demonstrate a progressive decline in performance for all classifiers when evaluated on samples from later periods. Whereas, the deep learning CNN model under the class abstraction maintains a certain stability over time. In comparison with eight state-of-the-art approaches, LongCGDroid achieves higher accuracy. [JJCIT 2023; 9(4.000): 328-346]

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]

For a set of graphs F , a graph is said to be F -free if it does not contain any graph in F as a subgraph. Let Ex s p (n , F) denote the graphs with the maximum spectral radius among all F -free graphs of order n. A linear forest is a graph whose connected components are paths. Denote by L s the family of all linear forests with s edges. In this paper the graphs in Ex s p (n , { K k + 1 , L s }) will be completely characterized when n is appropriately large. [ABSTRACT FROM AUTHOR]

The spectral radius ρ (A) is the maximum absolute value of the eigenvalues of a matrix A. In this paper, we establish some relationship between the spectral radius of a symmetric matrix and its principal submatrices, i.e., if A is partitioned as a 2 × 2 block matrix A = ( 0 A 12 A 21 A 22 ) , then ρ (A) 2 ≤ ρ 2 2 + θ ⁎ , where θ ⁎ is the largest real root of the equation μ 2 = (x − ν) 2 (ρ 2 2 + x) and ρ 2 = ρ (A 22) , μ = ρ (A 12 A 22 A 21) , ν = ρ (A 12 A 21). Furthermore, the results are used to obtain several upper bounds of the spectral radius of graphs, which strengthen or improve some known results. [ABSTRACT FROM AUTHOR]

A formula for the group inverse of trees, obtained recently, is shown to be applicable to a special class of weighted graphs G w studied here. Certain handpicked results, which hold for bipartite graphs, are shown to be true for this class. [ABSTRACT FROM AUTHOR]

Accurate prediction of traffic flow plays an important role in maintaining traffic order and traffic safety, which is a key task in the application of intelligent transportation systems (ITS). However, the urban road network has complex dynamic spatial correlation and nonlinear temporal correlation, and achieving accurate traffic flow prediction is a highly challenging task. Traditional methods use sensors deployed on roads to construct the spatial structure of the road network and capture spatial information by graph convolution. However, they ignore that the spatial correlation between nodes is dynamically changing, and using a fixed adjacency matrix cannot reflect the real road spatial structure. To overcome these limitations, this paper proposes a new spatial-temporal deep learning model: gated fusion adaptive graph neural network (GFAGNN). GFAGNN first extracts long-term dependencies on raw data through stacking expansion causal convolution, Then the spatial features of the dynamics are learned by adaptive graph attention network and adaptive graph convolutional network respectively, Finally the fused information is passed through a lightweight channel attention to extract temporal features. The experimental results on two public data sets show that our model can effectively capture the spatiotemporal correlation in traffic flow prediction. Compared with GWNET-conv model on METR-LA dataset, the three indexes in the 60-minute task prediction improved by 2.27%,2.06% and 2.13%, respectively. [ABSTRACT FROM AUTHOR]

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

Let G be a graph, and let $ A(G) $ A (G) be the adjacency matrix of G. The computation of permanent of $ A(G) $ A (G) is #p-complete. Computing permanent of $ A(G) $ A (G) is of great interest in quantum chemistry, statistical physics, among other disciplines. In this paper, we characterize the ordering of permanents of adjacency matrices of all graphs obtained from regular complete bipartite graph $ K_{p, p} $ K p , p by deleting six edges. As an application, we show that all graphs with a perfect matching obtained from $ K_{p, p} $ K p , p with six edges deleted are determined by their permanental polynomials. [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]

Epilepsy is a chronic brain disease, and identifying seizures based on electroencephalogram (EEG) signals would be conducive to implement interventions to help patients reduce impairment and improve quality of life. In this paper, we propose a classification algorithm to apply dynamical graph neural network with attention mechanism to single channel EEG signals. Empirical mode decomposition (EMD) are adopted to construct graphs and the optimal adjacency matrix is obtained by model optimization. A multilayer dynamic graph neural network with attention mechanism is proposed to learn more discriminative graph features. The MLP-pooling structure is proposed to fuse graph features. We performed 12 classification tasks on the epileptic EEG database of the University of Bonn, and experimental results showed that using 25 runs of ten-fold cross-validation produced the best classification results with an average of 99.83 % accuracy, 99.91 % specificity, 99.78 % sensitivity, 99.87 % precision, and 99.47 % F 1 score for the 12 classification tasks. [ABSTRACT FROM AUTHOR]

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

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]

The paper proposes a deep learning model based on Chebyshev Network Gated Recurrent Units, which is called Spectral Graph Convolution Recurrent Neural Network, for multichannel electroencephalogram emotion recognition. First, in this paper, an adjacency matrix capturing the local relationships among electroencephalogram channels is established based on the cosine similarity of the spatial locations of electroencephalogram electrodes. The training efficiency is improved by utilizing the computational speed of the cosine distance. This advantage enables our method to have the potential for real-time emotion recognition, allowing for fast and accurate emotion classification in real-time application scenarios. Secondly, the spatial and temporal dependence of the Spectral Graph Convolution Recurrent Neural Network for capturing electroencephalogram sequences is established based on the characteristics of the Chebyshev network and Gated Recurrent Units to extract the spatial and temporal features of electroencephalogram sequences. The proposed model was tested on the publicly accessible dataset DEAP. Its average recognition accuracy is 88%, 89.5%, and 89.7% for valence, arousal, and dominance, respectively. The experiment results demonstrated that the Spectral Graph Convolution Recurrent Neural Network method performed better than current models for electroencephalogram emotion identification. This model has broad applicability and holds potential for use in real-time emotion recognition scenarios. [ABSTRACT FROM AUTHOR]

We introduce a graph attached to mutually orthogonal Sudoku Latin squares. The spectra of the graphs obtained from finite fields are explicitly determined. As a corollary, we then use the eigenvalues to distinguish non-isomorphic Sudoku Latin squares. [ABSTRACT FROM AUTHOR]

For a simple graph G , the generalized adjacency matrix A α (G) is defined as A α (G) = α D (G) + (1 − α) A (G) , α ∈ [ 0 , 1 ] , where A (G) is the adjacency matrix and D (G) is the diagonal matrix of vertex degrees of G. This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of G. In this paper, we find the A α -spectrum of the joined union of graphs in terms of the spectrum of the adjacency matrices of its components and the zeros of the characteristic polynomials of an auxiliary matrix determined by the joined union. We determine the A α -spectrum of join of two regular graphs, the join of a regular graph with the union of two regular graphs of distinct degrees. As applications, we investigate the A α -spectrum of certain power graphs of finite groups. [ABSTRACT FROM AUTHOR]

The distance between an eigenvalue λ of a signed graph G ̇ and the spectrum of a signed graph H ̇ is defined as min { | λ − μ | : μ is an eigenvalue of H ̇ }. In this paper, we investigate this distance when H ̇ is a largest induced subgraph of G ̇ that does not have λ as an eigenvalue. We estimate the distance in terms of eigenvectors and structural parameters related to vertex degrees. For example, we show that | λ | | λ − μ | ≤ δ G ̇ ∖ H ̇ max { d H ̇ (i) d G ̇ − V (H ̇) (j) : i ∈ V (G ̇) ∖ V (H ̇) , j ∈ V (H ̇) , i ∼ j } , where δ G ̇ ∖ H ̇ is the minimum vertex degree in V (G ̇) ∖ V (H ̇). If H ̇ is obtained by deleting a single vertex i , this bound reduces to | λ | | λ − μ | ≤ d (i). We also consider the case in which λ is a simple eigenvalue and H ̇ is not necessarily a vertex-deleted subgraph, and the case when λ is the largest eigenvalue of an ordinary (unsigned) graph. Our results for signed graphs apply to ordinary graphs. They can be interesting in the context of eigenvalue distribution, eigenvalue location or spectral distances of (signed) graphs. [ABSTRACT FROM AUTHOR]

In this article we establish relationships between Leavitt path algebras, talented monoids and the adjacency matrices of the underlying graphs. We show that indeed the adjacency matrix generates in some sense the group action on the generators of the talented monoid. With the help of this, we deduce a form of the aperiodicity index of a graph via the talented monoid. We classify hereditary and saturated subsets via the adjacency matrix. This then translates to a correspondence between the composition series of the talented monoid and the so-called matrix composition series of the adjacency matrix. In addition, we discuss the number of cycles in a graph. In particular, we give an equivalent characterization of acyclic graphs via the adjacency matrix, the talented monoid and the Leavitt path algebra. Finally, we compute the number of linearly independent paths of certain length in the Leavitt path algebra via adjacency matrices. [ABSTRACT FROM AUTHOR]

This study aims to compare the longitudinal performance between machine-learning and deep-learning classifiers for Android malware detection, employing different levels of feature abstraction. Using a dataset of 200k Android apps labeled by date within a 10-year range (2013-2022), we propose the LongCGDroid, an image-based effective approach for Android malware detection. We use the semantic Call Graph API representation that is derived from the Control Flow Graph and Data Flow Graph to extract abstracted API calls. Thus, we evaluate the longitudinal performance of LongCGDroid against API changes. Different models are used; machine-learning models (LR, RF, KNN, SVM) and deep-learning models (CNN, RNN). Empirical experiments demonstrate a progressive decline in performance for all classifiers when evaluated on samples from later periods. However, the deep-learning CNN model under the class abstraction maintains a certain stability over time. In comparison with eight state-of-the-art approaches, LongCGDroid achieves higher accuracy. [ABSTRACT FROM AUTHOR]

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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]

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]

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 𝒟2n.

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

We consider particular weighted directed graphs with edges having colour red, blue or green such that each red edge has weight 1, each blue edge has weight $ -1 $ − 1 and each green edge has weight $ \mathrm {i} $ i (the imaginary unit). Such a directed graph is called a 3-coloured digraph (for short, a 3-CD). Every mixed graph and every signed graph can be interpreted as a 3-CD. We first study some structural properties of 3-CDs by means of the eigenvalues of the adjacency matrix. In particular, we give spectral criteria for singularity of such digraphs. Second, we consider weight-symmetric 3-CDs, i.e. those 3-CDs that are switching isomorphic to their negation. It follows that the class of weight-symmetric 3-CDs is included in the class of 3-CDs whose spectrum is symmetric (with respect to the origin). We give some basic properties and several constructions of weight-symmetric 3-CDs and establish constructions of 3-CDs which have symmetric spectrum but are not weight-symmetric. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field F p. We apply symmetric (row and column) elementary transformations to the adjacency matrix over F p in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over F p for any odd prime p. Our key tool is quadratic residues of F p. [ABSTRACT FROM AUTHOR]

Currently, there are two main limitations in single-label text classification tasks based on hierarchical graph attention networks. First, it cannot effectively extract text features. Second, there are few studies that highlight text features through the connection between text and labels. To address these two issues, a hierarchical graph attention network text classification model that integrates label information is proposed. The model constructs an adjacency matrix based on the relevance between sentence keywords and topics, and then uses word-level graph attention network to obtain vector representations of sentences. This method is based on randomly initialized target vectors and utilizes maximum pooling to extract specific target vectors for sentences, making the obtained sentence vectors have more obvious category features. After the word-level graph attention layer, a sentence-level graph attention network is used to obtain new text representations with word weight information, and pooling layers are used to obtain feature information for the text. On the other hand, GloVe pre-trained word vectors are used to initialize vector representations for all text labels, which are then interacted and fused with the feature information of the text to reduce the loss of original features, obtaining feature representations that are distinct from different texts. Experimental results on five public datasets (R52, R8, 20NG, Ohsumed, and MR) show that the classification accuracy of the model significantly exceeds other mainstream baseline models. [ABSTRACT FROM AUTHOR]

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

In this paper, we prove that every simple connected graph with α > n 2 has 1 as a normalized Laplacian eigenvalue with multiplicity at least 2 α − n , where n and α are the order and the independence number of the graph, respectively. Then we investigate graphs with α ≤ n 2 and having or not having 1 as a normalized Laplacian eigenvalue. [ABSTRACT FROM AUTHOR]

For a simple graph G, the ABC-index is a degree based topological index and is defined as where dv is the degree of the vertex υ in G. Recently, the Laplacian ABC-matrix was introduced in [22] is defined by where is the diagonal matrix of ABC-degrees and Ã(G) is the ABC-matrix of G: The eigenvalues of the matrix are called the Laplacian ABC-eigenvalues of G. In the article, we consider the problem of characterization of connected graphs having exactly three distinct Laplacian ABC-eigenvalues. We solve this problem for bipartite graphs, multipartite graphs, unicyclic graphs, regular graphs and prove the non-existence of such graphs with diameter greater than 2. We introduce the concept of trace norm of the matrix called the Laplacian ABC-energy of G. We obtain some upper and lower bounds for the Laplacian ABC-energy and characterize the extremal graphs which attain these bounds. [ABSTRACT FROM AUTHOR]