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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]

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]

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]

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]

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

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

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

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

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

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

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

Let Gw be a simple weighted graph with adjacency matrix A(Gw). The set of all eigenvalues of A(Gw) is called the spectrum of weighted graph Gw denoted by σ(Gw). The reciprocal eigenvalue property (or property R) for a connected weighted nonsingular graph Gw is defined as, if η ∈ σ(Gw) then 1 η ∈ σ(Gw). Further, if η and 1 η have the same multiplicities for each η ∈ σ(Gw) then this graph is said to have strong reciprocal eigenvalue property (or property SR). Similarly, a connected weighted nonsingular graph Gw is said to have anti-reciprocal eigenvalue property (or property -R) if η ∈ σ(Gw) then - 1 η ∈ σ(Gw). Furthermore, if η and - 1 η have the same multiplicities for each η ∈ σ(Gw) then strong anti-reciprocal eigenvalue property (or property -SR) holds for the weighted graph Gw. In this article, classes of weighted noncorona graphs satisfying property R and property -SR are studied. [ABSTRACT FROM AUTHOR]

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

Understanding the formation of feeding links provides insights into processes underlying food webs. Generally, predators feed on prey within a certain body‐size range, but a systematic quantification of such feeding niches is lacking. We developed a size‐constrained feeding‐niche (SCFN) model and parameterized it with information on both realized and non‐realized feeding links in 72 aquatic and 65 terrestrial food webs. Our analyses revealed profound differences in feeding niches between aquatic and terrestrial predators and variation along a temperature gradient. Specifically, the predator–prey body‐size ratio and the range in prey sizes increase with the size of aquatic predators, whereas they are nearly constant across gradients in terrestrial predator size. Overall, our SCFN model well reproduces the feeding relationships and predation architecture across 137 natural food webs (including 3878 species and 136,839 realized links). Our results illuminate the organisation of natural food webs and enables novel trait‐based and environment‐explicit modelling approaches. [ABSTRACT FROM AUTHOR]

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

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

We consider signed graphs with just 2 or 3 distinct eigenvalues, in particular (i) those with at least one simple eigenvalue, and (ii) those with vertex-deleted subgraphs which themselves have at most 3 distinct eigenvalues. We also construct new examples using weighing matrices and symmetric 3-class association schemes. [ABSTRACT FROM AUTHOR]

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

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

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

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

Given a finite group G and the subgroups lattice L(G) of G, the \textit{non--permutability graph of subgroups} ΓL(G) is introduced as the graph with vertices in L(G)\CL(G)(L(G)), where CL(G)(L(G)) is the smallest sublattice of L(G) containing all permutable subgroups of G, and edges obtained by joining two vertices X,Y if XY≠YX. Here we study the behaviour of the non-permutability graph of subgroups using algebraic properties of associated matrices such as the adjacency and the Laplacian matrix. Further, we study the structure of some classes of groups whose non-permutability graph is strongly regular. [ABSTRACT FROM AUTHOR]

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

Graph theory is one of the neglected branches of mathematics in Nigeria but with the most applications in other fields of research. This article shows the paucity, importance, and necessity of graph theory in the development of Nigeria. The adjacency matrix and dual graph of the Nigeria map were presented. The graph spectrum and energies (graph energy and Laplacian energy) of the dual graph were computed. Then the chromatic number, maximum degree, minimum spanning tree, graph radius, and diameter, the Eulerian circuit and Hamiltonian paths from the dual graph were obtained and discussed. [ABSTRACT FROM AUTHOR]

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

Power forecast for each renewable power plant (RPP) in the renewable energy clusters is essential. Though existing graph neural networks (GNN)-based models achieve satisfactory prediction performance by capturing dependencies among distinct RPPs, the static graph structure employed in these models ignores crucial lead-lag correlations among RPPs, resulting from the time difference of the air flow at spatially dispersed RPPs. To address this problem, this paper proposes a novel dynamic graph structure using multiple temporal granularity groups (TGGs) to characterize the lead-lag correlations among RPPs. A granular-based GNN called GGNet is designed to generate an optimal adjacency matrix for the proposed graph structure. Specifically, a two-dimensional convolutional neural network (2D-CNN) is used to quantify the uncertain lead-lag correlations among RPPs; secondly, a gate mechanism is used to calculate a dynamic adjacency matrix; Finally, a graph attention network (GAT) is used to aggregate the information on RPPs based on the well-learned adjacency matrix. Case studies conducted using real-world datasets, with wind power plants and photovoltaic power plants, show our method outperforms state-of-the-art (SoTA) ones with better performance. Compared with the SoTA models, the RMSE N and MAE N of wind power plants for 1–4 h forecast steps decreased on average by 22.925% and 13.18%, respectively; the RMSE N and MAE N of photovoltaic power plants for 1–4 h forecast steps decreased on average by 48.95% and 18.75%, respectively. The results show that the proposed framework can generate improved performance with accuracy and robustness. • Dynamic graph structure can clarify lead-lag power correlation of renewable plants. • A novel model can quantity the lead-lag power correlation of renewable plants. • Memory cell can make the adjacency matrix among renewable plants learnable. • A graph attention network can improve power forecast accuracy of renewable plants. [ABSTRACT FROM AUTHOR]

Ergonomic risk assessment (ERA) is commonly used to identify and analyze postures that are detrimental to the health of workers in industrial workplaces, which is vital to prevent work-related musculoskeletal disorders (WMSDs). Among the automatic approaches, algorithms based on graph convolutional networks (GCNs) have shown promising results in ERA using skeleton sequence as input. However, previous GCN-based methods still have certain limitations. First, the separated modeling of spatial and temporal information and the manually pre-defined topology of graph may restrict the representation diversity of the networks. Additionally, RNN-based temporal modeling often incurs high computational costs and fails to capture long-range temporal dependencies, thereby reducing flexibility in describing long videos. To overcome these challenges, in this study, we propose an attention-based adaptive spatial–temporal graph convolutional network (AAST-GCN), aiming to achieve effective and efficient action representation for ERA in long video. First, we employ an alternate modeling strategy to effectively capture the spatial–temporal information, and propose an improved adaptive adjacency matrix scheme to learn various coordination and relations of body-joints, thus enhancing the flexibility to model diverse postures. Furthermore, we introduce an efficient multi-scale temporal convolutional network as a replacement for RNN-based algorithms, enabling the network to extract various granularities of temporal features. Moreover, to make the network focuses on more valuable information, we employ a spatial–temporal interaction attention (STIA) module. Finally, the aforementioned modules are aggregated within a multi-task learning framework, with the action segmentation serving as the auxiliary task to further improve the accuracy of ERA. We conducted the ergonomic risk assessment on the UW-IOM and TUM Kitchen datasets using our network. Extensive experiments conducted on the most popular datasets UW-IOM and TUM Kitchen demonstrated that our proposed AAST-GCN outperforms other GCN-based methods. Ablation studies and visualization also prove the effectiveness of the individual sub-modules. [ABSTRACT FROM AUTHOR]

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

Let G be a graph with n vertices and m edges. The minimum edge dominating energy is defined as the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of the graph G. In this paper, some lower and upper bounds for the minimum edge dominating energy of graph G are established. [ABSTRACT FROM AUTHOR]

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

We consider signed graphs Ġ whose spectra are comprised of exactly two (distinct) eigenvalues that differ only in sign, abbreviated to signed graphs with two symmetric eigenvalues. We obtain some relationships between such signed graphs and their star complements. Our results include structural examinations and constructions of infinite families of signed graphs with two symmetric eigenvalues. We also determine the bases for the eigenspaces of the eigenvalues of Ġ in terms of the eigenspaces of its star complement. In particular, we consider the case in which a star complement has two symmetric eigenvalues, as well. [ABSTRACT FROM AUTHOR]

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

If A(G) and D(G) are respectively the adjacency matrix and the diagonal matrix of vertex degrees of a connected graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G) = α D(G) + (1 - α) A(G), where 0 ≤ α ≤ 1. The Aα (or generalized) spectral radius λ(Aα(G)) (or simply λα) is the largest eigenvalue of Aα(G). In this paper, we show that λα ≤ α Δ + (1 - α)√ 2m ( 1 - 1/ω), where m, Δ and ω = ω(G) are respectively the size, the largest degree and the clique number of G. Further, if G has order n, then we show that ... where di and mi are respectively the degree and the average 2-degree of the vertex Vi. [ABSTRACT FROM AUTHOR]

Multifactor component-amount designs are introduced. A method for the construction of multifactor component-amount design is devised. The method uses the kronecker product of matrices obtained from a single factor D-optimal component-amount design and an adjacency/incidence matrix of an undirected graph over a field 2 F. Further G-efficiency of designs is computed for two and three factors quadratic mixture component-amount models. This method can be used equally when factors have same or different number of components. [ABSTRACT FROM AUTHOR]

The nullity of a graph G is the multiplicity of zero as an eigenvalue of its adjacency matrix. An assignment of weights to the vertices of a graph, that satisfies a zero sum condition over the neighbors of each vertex, and uses maximum number of independent variables is denoted by a high zero sum weighting of the graph. This applicable tool is used to determine the nullity of the graph. Two types of graphs are defined, and the change of their nullities is studied, namely, the graph G+ab constructed from G by adding a new vertex ab which is joint to all neighbors of both vertices a and b of G, and G•ab which is obtained from G+ab by removing both vertices a and b. [ABSTRACT FROM AUTHOR]

A vertex u ∈ V1 in a bipartite graph G(V1∪V2;E) is redundant if all the vertices of V2 that are adjacent to u are also adjacent to a vertex w (≠= u) in V1. In other words, NG(u) ⊆ NG(w). Such vertices increase the cost of the network (when it is a communication network) or increase the unnecessary membership of the network (when it is a social network). An ideal cost effective network is the one where there is no redundant vertex. In this article, we model the above type of networks using graphs and call them antichain graphs. We characterize such graphs and study their properties. We show that if G and H are antichain graphs then so is their cartesian product G x H. We design few more methods to construct new antichain graphs from the existing ones. We also present generating procedures, which generate some regular and biregular antichain graphs. We define a critical edge with reference to the antichain property. We also characterize the critical edge. [ABSTRACT FROM AUTHOR]

In this work, we first give a more general form of the binomial, Fibonomial, and balance-binomial graphs that is called generalized Fibonomial graph. We also argue the spectra of generalized Fibonomial graph. Next, we introduce a new type of graph on Jacobsthal numbers that is called Jacobsthal-binomial graph and denoted by JBn . We obtain the adjacency, Laplacian and signless Laplacian characteristic polynomials of JBn, respectively. We lastly give inequalities for the adjacency, Laplacian and signless Laplacian energies of JBn. [ABSTRACT FROM AUTHOR]

We consider the notion of χ -completeness of a locally finite graph and we extend this notion to the weighted magnetic graph. Moreover, we establish a link between the magnetic adjacency matrix on line graph and the magnetic discrete Laplacian on 1-forms. [ABSTRACT FROM AUTHOR]

Statistical methods relating tensor predictors to scalar outcomes in a regression model generally vectorize the tensor predictor and estimate the coefficients of its entries employing some form of regularization, use summaries of the tensor covariate, or use a low dimensional approximation of the coecient tensor. However, low rank approximations of the coefficient tensor can suer if the true rank is not small. We propose a tensor regression framework which assumes a soft version of the parallel factors (PARAFAC) approximation. In contrast to classic PARAFAC where each entry of the coefficient tensor is the sum of products of rowspeci contributions across the tensor modes, the soft tensor regression (Softer) framework allows the row-specic contributions to vary around an overall mean. We follow a Bayesian approach to inference, and show that softening the PARAFAC increases model flexibility, leads to improved estimation of coefficient tensors, more accurate identication of important predictor entries, and more precise predictions, even for a low approximation rank. From a theoretical perspective, we show that employing Softer leads to a weakly consistent posterior distribution of the coefficient tensor, irrespective of the true or approximation tensor rank, a result that is not true when employing the classic PARAFAC for tensor regression. In the context of our motivating application, we adapt Softer to symmetric and semi-symmetric tensor predictors and analyze the relationship between brain network characteristics and human traits. [ABSTRACT FROM AUTHOR]

Graph embeddings, a class of dimensionality reduction techniques designed for relational data, have proven useful in exploring and modeling network structure. Most dimensionality reduction methods allow out-of-sample extensions, by which an embedding can be applied to observations not present in the training set. Applied to graphs, the out-of-sample extension problem concerns how to compute the embedding of a vertex that is added to the graph after an embedding has already been computed. In this paper, we consider the out-of-sample extension problem for two graph embedding procedures: the adjacency spectral embedding and the Laplacian spectral embedding. In both cases, we prove that when the underlying graph is generated according to a latent space model called the random dot product graph, which includes the popular stochastic block model as a special case, an out-of-sample extension based on a least-squares objective obeys a central limit theorem. In addition, we prove a concentration inequality for the out-of-sample extension of the adjacency spectral embedding based on a maximum-likelihood objective. Our results also yield a convenient framework in which to analyze trade-o between estimation accuracy and computational expenses, which we explore briefly. Finally, we explore the performance of these out-of-sample extensions as applied to both simulated and real-world data. We observe significant computational savings with minimal losses to the quality of the learned embeddings, in keeping with our theoretical results. [ABSTRACT FROM AUTHOR]

In this research paper we begin with study of family of n dimensional hyper cube graphs and establish some properties related to their distance, spectra, and multiplicities and associated eigen vectors and extend to bipartite double graphs[11]. In a more involved way since no complete characterization was available with experiential results in several inter connection networks on this spectra our work will add an element to existing theory. [ABSTRACT FROM AUTHOR]

For 0 ≤ α ≤ 1, Nikiforov proposed to study the spectral properties of the family of matrices Aα(G) = αD(G)+(1 - α)A(G) of a graph G, where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix of G. The α-spectral radius of G is the largest eigenvalue of Aα(G). For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove results concerning the behavior of the α-spectral radius under relocation of a pendant edge in a pendant path. We give upper bounds for the α-spectral radius for unicyclic graphs G with maximum degree ∆ ≥ 2, connected irregular graphs with given maximum degree and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with the second largest α-spectral radius among trees, and the unique tree with the largest α-spectral radius among trees with given diameter. We also determine the unique graphs so that the difference between the maximum degree and the α-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively. [ABSTRACT FROM AUTHOR]

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

It is a well-known fact that a graph of diameter d has at least d + 1 eigenvalues. A graph is d -extremal, if it has diameter d and exactly d + 1 eigenvalues. A graph is split if its vertex set can be partitioned into a clique and a stable set. Such graphs have diameter at most 3. We obtain a complete classification of the connected bidegreed 3-extremal split graphs using the association of split graphs with combinatorial designs. We also construct certain families of non-bidegreed 3-extremal split graphs. [ABSTRACT FROM AUTHOR]

In this paper our focus is on regular signed graphs with exactly 3 (distinct) eigenvalues. We establish certain basic results; for example, we show that they are walk-regular. We also give some constructions and determine all the signed graphs with 3 eigenvalues, under the constraint that they are either signed line graphs or have vertex degree 3. We also report our result of computer search on those with at most 10 vertices. [ABSTRACT FROM AUTHOR]

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