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A combinatorial approach is given to compute bases for eigenspaces of zero-divisor graphs of finite Boolean rings. A commutative monoid $ \mathcal {S} $ S of graphs is shown to contain a cyclic submonoid $ \mathcal {U} $ U that determines values of the entries of basis elements, while the members of its complement $ \mathcal {S}\setminus \mathcal {U} $ S ∖ U encode the supports of these elements. Furthermore, every member of $ \mathcal {S}\setminus \mathcal {U} $ S ∖ U is associated with a Catalan-triangle number, which counts the number of basis elements whose supports are determined by the given member. This is established by using a combinatorial interpretation of Catalan-triangle numbers to produce linearly independent sets of eigenvectors. [ABSTRACT FROM AUTHOR]

Let Γ be a simple finite graph with vertex set V (Γ) and edge set E (Γ) . Let R be an equivalence relation on V (Γ) . The R -super Γ graph Γ R is a simple graph with vertex set V (Γ) and two distinct vertices are adjacent if either they are in the same R -equivalence class or there are elements in their respective R -equivalence classes that are adjacent in the original graph Γ . We first show that Γ R is a generalized join of some complete graphs and using this we obtain the adjacency and Laplacian spectrum of conjugacy super commuting graphs and order super commuting graphs of dihedral group D 2 n (n ≥ 3) , generalized quaternion group Q 4 m (m ≥ 2) and the nonabelian group Z p ⋊ Z q of order pq, where p and q are distinct primes with q | (p - 1) . [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]

It is shown that the nilpotency of submatrices of a certain class of adjacency matrices is equivalent to the Collatz conjecture. Our main result extends the previous work of Alves et al. and clarifies a conjecture made by Cardon and Tuckfield. [ABSTRACT FROM AUTHOR]

A graph is claw-free if it does not contain a star of order 4 as an induced subgraph. In this paper, we characterize all claw-free connected graphs with all but four eigenvalues equal to 0 or − 1. [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]

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]

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

In light of the successful investigation of the adjacency matrix, a significant amount of its modification is observed employing numerous topological indices. The matrix corresponding to the well-known first Zagreb index is one of them. The entries of the first Zagreb matrix are d u i + d u j , if u i is connected to u j ; 0, otherwise, where d u i is degree of i-th vertex. The current work is concerned with the mathematical properties and chemical significance of the spectral radius ( ρ 1 ) associated with this matrix. The lower and upper bounds of ρ 1 are computed with characterizing extremal graphs for the class of unicyclic graphs and trees. The chemical connection of the first Zagreb spectral radius is established by exploring its role as a structural descriptor of molecules. The isomer discrimination ability of ρ 1 is also explained. [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]

Let G be a connected graph with adjacency matrix $ A(G) $ A (G) and distance matrix $ \mathcal {D}(G) $ D (G). The adjacency-distance matrix of G is defined as $ S(G) = \mathcal {D}(G) + A(G) $ S (G) = D (G) + A (G). In this paper, $ S(G) $ S (G) is generalized by the convex linear combinations \[ S_{\alpha}(G)=\alpha \mathcal{D}(G)+(1-\alpha)A(G) \] S α (G) = α D (G) + (1 − α) A (G) where $ \alpha \in [0,1] $ α ∈ [ 0 , 1 ]. Let $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) be the spectral radius of $ S_{\alpha }(G) $ S α (G). This paper presents results on $ S_{\alpha }(G) $ S α (G) with emphasis on $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) and some results on $ S(G) $ S (G) are extended to all α in some subintervals of $ [0,1] $ [ 0 , 1 ]. For $ \alpha \in [1/2,1] $ α ∈ [ 1 / 2 , 1 ] , the trees attaining the largest and the smallest $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) among trees of fixed order are determined and it is proved that $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) is a branching index. Moreover, for $ \alpha \in (1/2,1] $ α ∈ (1 / 2 , 1 ] , the graphs that uniquely minimize $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) : among all connected graphs of fixed order and fixed connectivity, and among all connected graphs of fixed order and fixed chromatic number are characterized. [ABSTRACT FROM AUTHOR]

We investigate the Sombor spectral properties of threshold graphs, a formula for the Sombor index is presented, the Sombor eigenvalues are given, graphs with simple Sombor eigenvalues are characterized, bounds on the smallest/largest Sombor eigenvalues are presented, the multiplicities of the Sombor eigenvalues are discussed, formulae for trace and determinant of the associated quotient matrix are given, the Sombor spread bound and the bounds on the Sombor energy along with the characterization of extremal graphs. At the end, the conclusion states that all our results are valid for adjacency matrix and other adjacency type matrices. [ABSTRACT FROM AUTHOR]

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

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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]

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. In this paper, two new classes of trees T (i , j) • T (p , q) • T (r , m , t) and K 1 , s • T (i , j) • T (p , q) • T (r , m , t) of diameter 6 are defined. We obtain their characteristic polynomials and give the necessary and sufficient conditions for them to be integral. We also present some sufficient conditions of such trees to be integral by computer search. We also prove that the problem of finding integral trees of diameter 6 is equivalent to the problem of solving some Diophantine equations. Finally, we propose two basic open problems about integral trees of diameter 6 for further study. [ABSTRACT FROM AUTHOR]

The classical Pauli group can be obtained as the central product of the dihedral group of 8 elements with the cyclic group of order 4. Inspired by this characterization, we introduce the notion of central product of Cayley graphs, which allows to regard the Cayley graph of a central product of groups as a quotient of the Cartesian product of the Cayley graphs of the factor groups. We focus our attention on the Cayley graph C a y (P n , S P n ) of the generalized Pauli group P n on n -qubits; in fact, P n may be decomposed as the central product of finite 2-groups, and a suitable choice of the generating set S P n allows us to recognize the structure of central product of graphs in C a y (P n , S P n ). Using this approach, we are able to recursively construct the adjacency matrix of C a y (P n , S P n ) for each n ≥ 1 , and to explicitly describe its spectrum and the associated eigenvectors. It turns out that C a y (P n , S P n ) is a (3 n + 2) -regular bipartite graph on 4 n + 1 vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for P 1 the three classical Pauli matrices, one gets the so-called Möbius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral. [ABSTRACT FROM AUTHOR]

In 2010, Butler introduced the unfolding operation on a bipartite graph to produce two bipartite graphs, which are cospectral for the adjacency and the normalized Laplacian matrices. In this article, we describe how the idea of unfolding a bipartite graph with respect to another bipartite graph can be extended to nonbipartite graphs. In particular, we describe how unfoldings involving reflexive bipartite, semi-reflexive bipartite, and multipartite graphs are used to obtain cospectral nonisomorphic graphs for the adjacency matrix. [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.

Aiming at the problem that existing image text cross-modal retrieval is difficult to fully exploit the local consistency in the mode in the common subspace, a cross-modal retrieval method based on improved graph convolution is proposed. In order to improve the local consistency within each mode, the modal diagram is constructed with a single sample as a node, fully mining the interactive information between features. In order to solve the problem that graph convolution network can only do shallow learning, the method of adding initial residual link and weight identity map in each layer of graph convolution is adopted to alleviate this phenomenon. In order to jointly update the central node features through higher-order and lower-order neighbor information, an improvement is proposed to reduce neighbor nodes and increase the number of layers in graph convolution network. In order to learn highly locally consistent and semantically consistent public representation, it shares the weights of common representation learning layer, and jointly optimizes the semantic constraints within the modes and the modal invariant constraints between modes in the common subspace. The experimental results show that on the two cross-modal data sets of Wikipedia and Pascal sentence, the average mAP values of different retrieval tasks are 2.2%~42.1% and 3.0%~54.0% higher than the 11 existing methods. [ABSTRACT FROM AUTHOR]

Let R be the commutative ring R = Zp²[x]/〈x²〉 with identity and Z*(R) be the set of all non-zero zero-divisors of R. Then, (R) is said to be a zero-divisor graph if and only if a · b = 0 where a; b 2 V (Γ(R)) = Z*(R) and (a; b) 2 E(Γ(R)). Let λ1, λ2,... λ2 be the eigenvalues of the adjacency matrix, and let µ1, µ2,..., µn be the eigenvalues of the Laplacian matrix of Γ(R). Then we discuss the energy E(Γ(R)) = Pn i=1jij and the Laplacian energy LE(Γ(R)) = Pn i=1 λi Γ 2m n where n and m are the order and size of Γ(R). [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]

This work generalizes the recently introduced bubble entropy (BE) algorithm for univariate time series to graph and network analysis. In this paper, we introduce a modified method, called bubble entropy for graph signals (BEG), as an invaluable tool for detecting the irregularity of signals defined on graphs. Our algorithm is based on using the adjacency matrix to combine the signal values with the topology of the graph. Experiments on both synthetic and real data demonstrate the availability of the proposed measures in detecting the irregularity of graph signals and identifying topological changes in graphs. [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 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]

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]

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]

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

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

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.

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 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]

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]

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]

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]

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]

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]

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]

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]

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]