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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]

Although graph convolutional networks have achieved good performances in skeleton‐graph‐based action recognition, there are still some problems which include the incomplete utilization of skeleton graph features and the lacking of logical adjacency information between nodes in adjacency matrix. In this article, a human action recognition algorithm is proposed based on multiple features from the skeleton graph to solve these problems. More specifically, an improved adjacency matrix is constructed to make full use of the multiple skeleton graph features. These features include local differential features, multi‐scale edge features, features of the original skeleton graph, nodal features, and nodal motion features. Extensive results are conducted on four standard datasets (NTU RGB‐D 60, NTU RGB‐D 120, Kinetics, and Northwestern‐UCLA). The experimental results show that the proposed algorithm outperforms the SOTA action recognition algorithms. [ABSTRACT FROM AUTHOR]

The most common manifestation of neurological disorders in children is the occurrence of epileptic seizures. In this study, we propose a multi-branch graph convolutional network (MGCNA) framework with a multi-head attention mechanism for detecting seizures in children. The MGCNA framework extracts effective and reliable features from high-dimensional data, particularly by exploring the relationships between EEG features and electrodes and considering the spatial and temporal dependencies in epileptic brains. This method incorporates three graph learning approaches to systematically assess the connectivity and synchronization of multi-channel EEG signals. The multi-branch graph convolutional network is employed to dynamically learn temporal correlations and spatial topological structures. Utilizing the multi-head attention mechanism to process multi-branch graph features further enhances the capability to handle local features. Experimental results demonstrate that the MGCNA exhibits superior performance on patient-specific and patient-independent experiments. Our end-to-end model for automatic detection of epileptic seizures could be employed to assist in clinical decision-making. [ABSTRACT FROM AUTHOR]

The graph spectra analyze the structure of the graph using eigenspectra. The spectral graph theory deals with the investigation of graphs in terms of the eigenspectrum. In this paper, the sequence of lower bounds for the spectral radius of digraph D having at least one doubly adjacent vertex in terms of indegree is proposed. Particularly, it is exhibited that ρ (D) ≥ α j = ∑ p = 1 m (χ j + 1 (p)) 2 ∑ p = 1 m (χ j (p)) 2 , such that equality is attained iff D = G ↔ + { DE ∉ Cycle}, where each component of associated graph is a k-regular or (k 1 , k 2) semiregular bipartite. By utilizing the sequence of lower bounds of the spectral radius of D , the sequence of upper bounds of energy of D , where the sequence decreases when e U ≤ α j and increases when e U > α j , are also proposed. All of the obtained inequalities are elaborated using examples. We also discuss the monotonicity of these sequences. [ABSTRACT FROM AUTHOR]

During automated manufacturing processes, multiple sensors are attached to facilities to collect and analyze analog data for detecting operational anomalies. However, owing to facility devices being interlinked by a control system, simultaneous examination of the control system and analog data enhances the accuracy of anomaly detection and diagnosis of root causes. We proposed a system detecting anomalies by integrating an internal control system with external analog data and representing it in a graph structure. The system generates and combines the adjacency and feature matrices for training a convolutional autoencoder model to identify operational anomalies. Performance tests revealed distinct operational patterns in the cycle data flagged by the model as anomalies. The system diagnosed the root cause of anomalies, such as control operation sequencing, timing variances, and shifts in analog or video signals. This approach may enhance the productivity and quality of the manufacturing processes by facilitating anomaly detection and cause diagnosis. [ABSTRACT FROM AUTHOR]

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]

In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too. [ABSTRACT FROM AUTHOR]

The increasingly widespread use of new power equipment has brought new disturbances to the power system and has placed increasing demands on power quality. In order to make full use of the power quality indicators in the national standards and to make a more comprehensive and integrated evaluation of power quality, this study proposes a power quality evaluation method based on graph convolutional network. A power quality assessment system with graded indicators is proposed according to the current national standards. The correlation between the various power quality assessment indicators is initially determined, and on this basis the indicator relationship diagram is determined, a graph neural network model is built and trained, and the error rate of the test set is 9.02%. A comparison and analysis with other assessment methods using actual measurement data of a power system proves that the proposed method is more effective in assessing power quality over a long time span. [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]

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]

Abstract Although graph convolutional networks have achieved good performances in skeleton‐graph‐based action recognition, there are still some problems which include the incomplete utilization of skeleton graph features and the lacking of logical adjacency information between nodes in adjacency matrix. In this article, a human action recognition algorithm is proposed based on multiple features from the skeleton graph to solve these problems. More specifically, an improved adjacency matrix is constructed to make full use of the multiple skeleton graph features. These features include local differential features, multi‐scale edge features, features of the original skeleton graph, nodal features, and nodal motion features. Extensive results are conducted on four standard datasets (NTU RGB‐D 60, NTU RGB‐D 120, Kinetics, and Northwestern‐UCLA). The experimental results show that the proposed algorithm outperforms the SOTA action recognition algorithms.

The evaluation of the agricultural land in the Republic of Moldova represents the main factor in the sustainable rural space development and it emphasizes economic criteria for the efficiency of land resources. The hedonic price in the sales-purchase transactions of parcels in agricultural land may be the scientific innovation and originality of the geographically weighted regression methodology with the spatial approach in the used econometric model. The price of agricultural land used parcels in Maiovca village, Ocnița district were evaluated by using the data from transactions registered in 2022 and the results of the geographically weighted regression stands up the suitable conclusions of the licensed assessors in the field. The autocorrelation coefficient value evaluated by SAR is ρ = 0.072 according to Cobb Douglas model confirms the high quality of the regression techniques carried out and the used spatial methodology can be implemented in cadastral practice for the taxation in the agrarian sector. The geographically weighted regression utility proposed in this methodology has advantage in research area because of the similar evaluation techniques carried out in the countries of the European Union that are adjacent to the Republic of Moldova.

In this article, we study a question of Hong from 1993 related to the minimum spectral radii of the adjacency matrices of connected graphs of given order and size. Hong asked if it is true that among all connected graphs of given number of vertices nn and number of edges ee, the graphs having minimum spectral radius (the minimizer graphs) must be almost regular, meaning that the difference between their maximum degree and their minimum degree is at most one. In this article, we answer Hong’s question positively for various values of nn and ee, and in several cases, we determined the graphs with minimum spectral radius.

Let G be a graph of size m and [Formula: see text] be the spectral radius of its adjacency matrix. A graph is said to be H-free if it does not contain a subgraph isomorphic to H. Let [Formula: see text] be the graph obtained by adding a pendent vertex to a cycle of length l. In this paper, we determine the graphs with the maximum spectral radius over [Formula: see text]-free graphs, and [Formula: see text]-free graphs with [Formula: see text].

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]

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.

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]

As digital communication becomes an integral part of daily life, the rise in cybercrime has become increasingly concerning. To safeguard messages from hacking and tampering, concealing original information is essential. Modern cryptography goes beyond simple message encryption, emphasizing the protection of encrypted data against cyber attacks. Besides enabling secure communication, modern cryptography also prioritizes message authentication and integrity. This paper presents a symmetric block cipher model that utilizes graph structures. The encryption process in this method occurs in two stages. In the first stage, the message is encrypted using the adjacency matrix of an undirected graph. The second stage then applies an exclusive OR (XOR) operation to the output from the first stage. Unique encryption and decryption keys are generated for each block and each round, derived through the Minimum Spanning Tree (MST) technique on an undirected graph. This approach enables multiple peers to exchange messages simultaneously. One of the participants selects random ASCII characters, equal in number to the block length, and uploads these to the cloud. All group members then generate block keys using a shared, confidential key generation technique. [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]

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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]

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]

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]

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]

The most common manifestation of neurological disorders in children is the occurrence of epileptic seizures. In this study, we propose a multi-branch graph convolutional network (MGCNA) framework with a multi-head attention mechanism for detecting seizures in children. The MGCNA framework extracts effective and reliable features from high-dimensional data, particularly by exploring the relationships between EEG features and electrodes and considering the spatial and temporal dependencies in epileptic brains. This method incorporates three graph learning approaches to systematically assess the connectivity and synchronization of multi-channel EEG signals. The multi-branch graph convolutional network is employed to dynamically learn temporal correlations and spatial topological structures. Utilizing the multi-head attention mechanism to process multi-branch graph features further enhances the capability to handle local features. Experimental results demonstrate that the MGCNA exhibits superior performance on patient-specific and patient-independent experiments. Our end-to-end model for automatic detection of epileptic seizures could be employed to assist in clinical decision-making.

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.

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]