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

Let G be a connected graph. Denote by d i the degree of a vertex v i in G. Let f (x , y) > 0 be a real symmetric function. Consider an edge-weighted graph in such a way that for each edge v i v j of G, the weight of v i v j is equal to the value f (d i , d j) . Therefore, we have a degree-based weighted adjacency matrix A f (G) of G, in which the (i, j)-entry is equal to f (d i , d j) if v i v j is an edge of G and is equal to zero otherwise. Let x be a positive eigenvector corresponding to the largest eigenvalue λ 1 (A f (G)) of the weighted adjacency matrix A f (G) . In this paper, we first consider the unimodality of the eigenvector x on an induced path of G. Second, if f(x, y) is increasing in the variable x, then we investigate how the largest weighted adjacency eigenvalue λ 1 (A f (G)) changes when G is perturbed by vertex contraction or edge subdivision. The aim of this paper is to unify the study of spectral properties for the degree-based weighted adjacency matrices of graphs. [ABSTRACT FROM AUTHOR]

Exploring a general and effective method of isomorphism identification is an arduous task. For this aim, some new concepts such as binary link path and link-assortment adjacency matrix are introduced in this paper. On this basis, a new method is proposed to improve the operability of isomorphism identification and relieve the computational pressure. By generating new elements from structural features and connection relations among links or vertices in the graph, it transforms traditional high-ranking adjacency matrix into new low-ranking adjacency matrix. Some typical graphs including kinematic chains, topological graphs, and planetary gear trains are used to verify the effectiveness of the proposed method. As shown by a comparison with results in the cited references, the proposed method is available in isomorphism identification. [ABSTRACT FROM AUTHOR]

The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest in this question dates back to the early 1980 s, when P. J. Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable adjacency matrix. This was answered in the affirmative by Babai (J Graph Theory 9:363–370, 1985). Then Babai posed the open problems of constructing a 2-arc-transitive digraph and a vertex-primitive digraph whose adjacency matrices are not diagonalizable. In this paper, we solve Babai's problems by constructing an infinite family of s-arc-transitive digraphs for each integer s ≥ 2 , and an infinite family of vertex-primitive digraphs, both of whose adjacency matrices are non-diagonalizable. [ABSTRACT FROM AUTHOR]

Based on the extended adjacency matrix, the spectral radius and the energy of the extended adjacency matrix are found that they possess high discriminating power and correlate well with a number of physicochemical properties and biological activities of organic compounds. In this research, by establishing a relationship between directed graphs and undirected graphs, we mainly present the Sachs theorem and the Coulson’s integral formula of the energy of the extended adjacency matrix of G. By using the formula, we can compare the energies of the extended adjacency matrices of two graphs and we find that the graph energy is less than the extended energy in the case of a tree. [ABSTRACT FROM AUTHOR]

A quantum particle evolving by Schrödinger's equation contains, from the kinetic energy of the particle, a term in its Hamiltonian proportional to Laplace's operator. In discrete space, this is replaced by the discrete or graph Laplacian, which gives rise to a continuous-time quantum walk. Besides this natural definition, some quantum walk algorithms instead use the adjacency matrix to effect the walk. While this is equivalent to the Laplacian for regular graphs, it is different for non-regular graphs and is thus an inequivalent quantum walk. We algorithmically explore this distinction by analyzing search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix. The two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked vertices, and in what constitutes a natural initial state. Thus the choice of the Laplacian or adjacency matrix to effect the walk has important algorithmic consequences. [ABSTRACT FROM AUTHOR]

Graph convolutional networks (GCNs) have become the de facto approaches and achieved state-of-the-art results for circumventing many real-world problems on graph-structured data. However, these networks are usually shallow due to the over-smoothing of GCNs with many layers, which limits the expressive power of learning graph representations. The current methods of solving the limitations have the bottlenecks of high complexity and many parameters. Although Simple Graph Convolution (SGC) reduces the complexity and parameters, it fails to distinguish the feature information of neighboring nodes at different distances. To tackle the limits, we propose MulStepNET, a stronger multi-step graph convolutional network architecture, that can capture more global information, by simultaneously combining multi-step neighborhoods information. When compared to existing methods such as GCN and MixHop, MulStepNET aggregates neighborhoods information at more distant distances via multi-power adjacency matrix while fitting fewest parameters and being computationally more efficient. Experiments on citation networks including Pubmed, Cora, and Citeseer demonstrate that the proposed MulStepNET model improves over SGC by 2.8, 3.3, and 2.1% respectively while keeping similar stability, and achieves better performance in terms of accuracy and stability compared to other baselines. [ABSTRACT FROM AUTHOR]

The Estrada topological index EE, based on the eigenvalues of the adjacency matrix, is degenerate for cospectral graphs. By additionally considering the eigenvectors, two new topological indices are devised, which have reduced degeneracy for alkanes or cyclic graphs. Index RVa shows similarity to EE in ordering of alkanes with 8-10 carbon atoms, whereas index RVb is more similar to the average distance-based connectivity (Balaban index J). Inter-correlations between these four topological indices are discussed, indicating which factors have predominant influence. [ABSTRACT FROM AUTHOR]

Based on the characteristic polynomial coefficients (CPCs) of the adjacency matrix A′ of heterogeneous molecular graphs of the molecules containing a tetravalent heteroatom >Si< (>Ge<, >Sn<), etc. in the chain, a 13-constant additive scheme for the calculation of their physicochemical properties is obtained. The structural meaning of CPCs of the adjacency matrix A′ is established. By the formula obtained the formation enthalpies Δ H of SiCH alkylsilanes not studied experimentally are calculated. [ABSTRACT FROM AUTHOR]

There is substantial interest in the current literature – spanning finance, economics, engineering and medical imaging – on the relationship structure between several nodes in a complex system. In this paper, we extend the literature by developing model and inference for complex networks in terms of latent factors, by extracting the hidden factors that plays significant role in the configuration of inter node relationships. Together, we extend inferences to applications where the underlying network structure is also latent; that is, the adjacency matrix is unobserved. Here, we consider a Bayesian variant of the matrix factorization technique to develop a structural model of the latent adjacency matrix. There are many potential applications. For illustration, we consider a latent network of firms in the in the US automotive sector, where the central object is to understand the impact of an economic shock on firms (or nodes of the network). An important question centers around the factors that affect the stability and resilience of inter node relationships in a network. In the automotive sector application, we would like to know whether these relationship structures are driven by collaborative or competitive environments? What are the effects of a collaborative or competitive role played by a specific firm on the configuration of the relationship network? Using accounting data on firm sales and costs, we use our proposed object oriented factorization methodology to provide explanation of the estimated network links between 3 major US auto manufacturers and their intermediate suppliers. [ABSTRACT FROM AUTHOR]

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

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

Forecasting traffic flow is crucial for Intelligent Traffic Systems (ITS), traffic control, and traffic management systems. Complex spatial and temporal interactions of traffic networks make traffic forecasting tasks challenging. Recently, Graph Convolutional Network (GCN) has attracted researchers' attention as it can better represent graph-shaped road networks and extract spatial features of traffic. However, traditional GCN has some drawbacks since it uses a static adjacency matrix which is unable to capture the time-varying features of traffic propagation. To overcome this, we represent the traffic road network as a dynamic graph and use a probabilistic spatiotemporal adjacency matrix to identify the time-varying impacts of adjacent roads on target roads in GCN. In addition, to find the similarity among the nonadjacent nodes, we have employed node-specific learning in GCN rather than sharing parameters in traditional GCN. This node-specific learning helps our model to learn detailed characteristics of road networks. For temporal feature extractions, we used a Gated Recurrent Unit (GRU) that captures the local trend of traffic flow and an attention mechanism to capture the global trend of traffic flow. We compared the performance of our model with baseline models using two real-world datasets. Experimental results show that our model is effective in forecasting both short and long-term traffic flow. Source code of our model is available at https://github.com/atkia/PSTGCN [ABSTRACT FROM AUTHOR]

Topological indices are widely used for characterizing molecular graphs, establishing relationships between structure and properties of molecules. In this article we discuss adjacency matrix and three topological indices: Wiener index, Laplacian energy and Zagreb indices of zero-divisor graphs of Z n . We also provide a MATLAB code for Laplacian energy and Zagreb indices of Γ (Z n) . [ABSTRACT FROM AUTHOR]

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

Graph convolutional networks (GCNs) have achieved remarkable performance in the 2D-to-3D human pose estimation (HPE) task. The adjacency matrix in GCNs is crucial for feature aggregation in 3D HPE. However, existing GCN-based methods excessively rely on the fixed adjacency matrix to aggregate joint features from one-hop neighbor at a single scale, which limits the feature representation of skeleton data. To better improve the performance of 3D HPE, we have designed a multi-scale parallel adaptive graph network (MPA-GNet) for 3D HPE. The proposed network consists of three parallel multi-scale subgraph networks (PMS-Net) to efficiently capture human joint features at different scales. Specially, a multi-scale feature fusion module is devised to process multi-scale graph structural features and exchange information to generate rich hierarchical representations for skeleton data. To flexible construct graph topology in different scales, a special designed adaptive attention adjacency graph convolution network and a cluster graph pooling module are designed to construct the MPA-GNet in a parallel manner and capture the local subgraphs information in each PMS-Net. Finally, we conduct experiments on two 3D human pose challenging benchmark datasets Human3.6M and HumanEva-I for evaluating the effectiveness of the proposed model. The experimental results demonstrate that our model achieves competitive performance compared with some state-of-the-art 3D HPE methods. [ABSTRACT FROM AUTHOR]

Traffic flow prediction plays a critical role in urban traffic management and planning. Accurate prediction of traffic flow can enhance traffic efficiency, improve traffic safety, conserve traffic resources, and promote sustainable urban development. Graph convolutional networks (GCN) have great potential in traffic flow prediction due to their ability to handle complex data with topological structures while considering spatial relationships between nodes in data. However, it is difficult for the traditional GCN-based models to capture dynamic spatiotemporal dependencies between the stations since these models usually use static relationship graph in GCN. This paper presents an enhanced approach called the coupled generative graph convolution model (CGGCN) for capturing dynamic regional traffic relationships. The CGGCN learns the adjacency matrix of graph convolution in a generative manner, thereby improving the ability of previous graph convolution models to effectively capture the dynamic relationships of regional traffic within multi-layer structures. Additionally, the paper optimizes the fusion method of coupling graph convolution in multi-level matrix feature flow information aggregation, leading to the efficient extraction of spatiotemporal information. To validate the effectiveness of the proposed model, extensive experiments were conducted on two real traffic datasets obtained from New York, namely NYCBike and NYCTaxi. The results demonstrate the superior performance of the CGGCN model when compared to other eight baseline models. [ABSTRACT FROM AUTHOR]

Predicting olfactory perceptions from odorant molecules is challenging due to the complex and potentially discontinuous nature of the perceptual space for smells. In this study, we introduce a deep learning model, Mol-PECO (Molecular Representation by Positional Encoding of Coulomb Matrix), designed to predict olfactory perceptions based on molecular structures and electrostatics. Mol-PECO learns the efficient embedding of molecules by utilizing the Coulomb matrix, which encodes atomic coordinates and charges, as an alternative of the adjacency matrix and its Laplacian eigenfunctions as positional encoding of atoms. With a comprehensive dataset of odor molecules and descriptors, Mol-PECO outperforms traditional machine learning methods using molecular fingerprints and graph neural networks based on adjacency matrices. The learned embeddings by Mol-PECO effectively capture the odor space, enabling global clustering of descriptors and local retrieval of similar odorants. This work contributes to a deeper understanding of the olfactory sense and its mechanisms. [ABSTRACT FROM AUTHOR]

Context: Pattern metrics drawn from image processing and remote sensing have been applied as descriptors of the texture of landscape gradient data. Like some classical pattern metrics in ecology, texture has several facets which are measured by examining an adjacency matrix—the frequencies of co-occurring pixel values on a map—in different ways. Objectives: To improve the interpretation and application of such metrics in landscape ecology we reformulate and interpret several of them by analogy to traditional metrics used with categorical data. Results and conclusions: 1. Four of the eight classical texture metrics measure attraction—the tendency for the same or similar values to be adjacent. Four others measure dispersion—the diversity of adjacencies relative to the entire adjacency matrix, the diagonal of the matrix, or the origin of the matrix. 2. The attraction metrics (dissimilarity, contrast, inverse difference, and homogeneity) differ only in the algebraic weights applied to different parts of an adjacency matrix. 3. The dispersion metrics (entropy, uniformity, difference entropy, and sum entropy) can be made more comparable by rescaling them to their maximum possible values. 4. While the metrics may be applied to any adjacency matrix, the choices about the method used to create an adjacency matrix have subtle yet important implications for the use and comparability of some metrics. [ABSTRACT FROM AUTHOR]

In this paper, first we introduce the notions of size, order, and degree of vertices and edges in a soft graph, and we give some examples for a better understanding of these concepts. Then, we define the fundamental concepts of the adjacency matrix of soft graphs, complement of soft graphs, and regularity based on the degree of vertices of soft graphs, and we investigate some related results. Moreover, we introduce the concept of soft multiset and soft multigraph and define the notions of union, intersection, complement, and sum, and then we investigate the relations among them. Finally, we give an application of soft graphs in controlling urban traffic flows. [ABSTRACT FROM AUTHOR]

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Under elimination of two arcs, spectrum of digraph having ring structure and two Hamiltonian cycles is shown real, if and only if the number of vertices is even and “interval between arcs,” eliminated from the same cycle, is maximal. Possibility to apply the obtained results to evaluate failure-resistance of networks with ring topology is considered. [ABSTRACT FROM AUTHOR]

Subtractive Pixel Adjacency Matrix (SPAM) features perform well in detecting spatial-domain steganographic algorithm. Further, some methods of SPAM features can be applied to rich models and steganalysis based on deep learning. Therefore, this paper presents a study on SPAM features and it is divided into two parts: in the first part, impact of spatial-domain steganographic on difference between adjacent pixels is first analyzed. Then, three SPAM features are proposed with the same range of differences and different orders of Markov chain. Following that, the influences of order of Markov chain and range of differences on SPAM features are analyzed, and we find that detection accuracy of SPAM features increases with the range of differences increasing; in the second part, SPAM feature is first divided into several modules according to the conclusion. Then, taking detection accuracy of support vector machine (SVM) classifier and mutual information as metrics and module as a unit, a Novel Feature Selection (NFS) algorithm and an Improved Feature Selection algorithm are proposed. Experimental results show that the NFS algorithm can achieve higher detection accuracy than several existing algorithms. [ABSTRACT FROM AUTHOR]

This paper expands on the graph-theoretic content of my contributed talk at the Seventh International Conference on Clifford Algebras and Their Applications. A well-known result in graph theory states that when A is the adjacency matrix of a finite graph G, the entries of A k represent numbers of k-step walks existing in G. However, the adjacency matrix fails to distinguish between walks and “self-avoiding” walks (i.e., walks without repeated vertices). Utilizing elements of abelian, nilpotent-generated algebras, a “new” adjacency matrix is associated with a finite graph on n vertices. By considering entries of $${\mathcal{A}}^k$$ , where $${\mathcal{A}}$$ is an appropriate nilpotent adjacency matrix, one is able to recover the self-avoiding k-walks in any finite graph. In particular, a graph’s Hamiltonian cycles are enumerated by the top-form coefficient in the trace of $${\mathcal{A}}^n$$ when n is the number of vertices in the graph. By considering the ℓ th power of the trace of $${\mathcal{A}}^k$$ , ℓ-tuples of pairwise-disjoint k-cycles are recovered. By defining a nilpotent transition matrix associated with a time-homogeneous Markov chain, a method of computing probabilities of self-avoiding random walks on finite graphs is developed. Expected hitting times of specific states in Markov chains and expected times of first self-intersection of random walks are also recovered using these methods. The algebra used to define the nilpotent adjacency matrix of a graph on n vertices is not itself a Clifford algebra, but it can be constructed within the 2 n-particle fermion algebra $${\mathcal{C}}\ell_{2n,2n}$$ , indicating potential connections to quantum computing. [ABSTRACT FROM AUTHOR]

Flowcharts have broad applications in the fields of software development, engineering design, and scientific experimentation. Current flowchart data structure is mainly based on the adjacency list, cross-linked list, and adjacency matrix of the graph structure. Such design originated from the fact that any two nodes could have a connection relationship. But flowcharts have clear regularities, and their nodes have a certain inflow or outflow relationship. When graph structures such as an adjacency table or an adjacency matrix are used to store a flowchart, there is a large room for optimization in terms of traversal time and storage complexities, as well as usage convenience. In this paper we propose two hierarchical data structures for flowchart design. In the proposed structures, a flowchart is composed of levels, layers, and numbered nodes. The nodes between layers are connected according to a certain set of systematic design rules. Compared with the traditional graph data structures, the proposed schemes significantly reduce the storage space, improve the traversal efficiency, and resolve the problem of nesting between sub-charts. Experimental data based on flowchart examples used in this paper show that, compared with adjacency list, the hierarchical table data structure reduces the traversal time by 50% while their storage spaces are similar; compared with adjacency matrix, the hierarchical matrix data structure reduces the traversal time by nearly 70% and saves the storage space by about 50%. The proposed structures could have broad applications in flowchart-based software development, such as low-code engineering for smart industrial manufacturing. [ABSTRACT FROM AUTHOR]

In the wake of Dutta and Adhikari, who in 2020 used partial transposition in order to get pairs of cospectral graphs, we investigate partial transposition for Hermitian complex matrices. This allows us to construct infinite pairs of complex unit gain graphs (or T -gain graphs) sharing either the spectrum of the adjacency matrix or even the spectrum of all the generalized adjacency matrices. This investigation also sheds new light on the classical case, producing examples that were still missing even for graphs. Partial transposition requires a block structure of the matrix: we interpreted it as if coming from a composition of T -gain digraphs. By proposing a suitable definition of rigidity specifically for T -gain digraphs, we then produce the first examples of pairs of non-isomorphic graphs, signed graphs and T -gain graphs obtained via partial transposition of matrices whose blocks form families of commuting normal matrices. In some cases, the non-isomorphic graphs detected in this way turned out to be hardly distinguishable, since they share the adjacency, the Laplacian and the signless Laplacian spectrum, together with many non-spectral graph invariants. [ABSTRACT FROM AUTHOR]

Identifying the rank of species in a complex ecosystem is a difficult task, since the rank of each species invariably depends on the interactions stipulated with other species through the adjacency matrix of the network. A common ranking method in economic and ecological networks is to sort the nodes such that the layout of the reordered adjacency matrix looks maximally nested with all nonzero entries packed in the upper left corner, called Nestedness Maximization Problem (NMP). Here we solve this problem by defining a suitable cost-energy function for the NMP which reveals the equivalence between the NMP and the Quadratic Assignment Problem, one of the most important combinatorial optimization problems, and use statistical physics techniques to derive a set of self-consistent equations whose fixed point represents the optimal nodes' rankings in an arbitrary bipartite mutualistic network. Concurrently, we present an efficient algorithm to solve the NMP that outperforms state-of-the-art network-based metrics and genetic algorithms. Eventually, our theoretical framework may be easily generalized to study the relationship between ranking and network structure beyond pairwise interactions, e.g. in higher-order networks. Species forming complex ecological or economic ecosystems are organized in hierarchies and the ranks of such species are determined by the adjacency matrix of their interaction network. We introduce a framework to calculate the ranks of species by finding the optimal permutation of rows and columns that makes the adjacency matrix maximally nested. [ABSTRACT FROM AUTHOR]

Pose estimation in crowded scenes is key to understanding human behavior in real-life applications. Most existing CNN-based pose estimation methods often depend on the appearance of visible parts as cues to localize human joints. However, occlusion is typical in crowded scenes, and invisible body parts have no valid features for joint localization. Introducing prior information about the human pose structure to infer the locations of occluded parts is a natural solution to this problem. In this paper, we argue that learning structural information based on human joints alone is not enough to address human body variations and could be prone to overfitting. From a perspective on the human pose as a dual representation of joints and limbs, we propose a pose refinement network, coined as dual graph network (DGN), to jointly learn its structural information of body joints and limbs by incorporating the cooperative constraints between two branches. Specifically, our DGN has two coupled graph convolutional network (GCN) branches to model the structure information of joints and limbs. Each stage in the branch is composed of a feature aggregator and a GCN module for inter-branch information fusion and intra-branch context extraction, respectively. In addition, to enhance the modeling capacity of GCN, we design an adaptive GCN layer (AGL) embedded in the GCN module to handle each pose instance based on its graph structure. We also propose a heatmap-guided sampling to leverage the features of the body parts to provide rich visual features for the inference of occluded parts. We perform extensive experiments on five challenging datasets to demonstrate the effectiveness of our DGN on pose estimation. Our DGN obtains significant performance improvement from 67.9 to 72.4 mAP in the CrowdPose dataset with the same CNN-based pose estimator and training strategy as the OPEC-Net. It shows that, compared to the OPEC-Net only considering joints, our DGN has a clear advantage due to the joint consideration of both joints and limbs. Meanwhile, our DGN is also helpful for pose estimation in general datasets (i.e., COCO and Pose track) with less occlusion and mutual interference, demonstrating the generalization power of DGN on refining human poses. [ABSTRACT FROM AUTHOR]

For a simple graph G, let A(G) be the adjacency matrix and D(G) be the diagonal matrix of the vertex degrees of graph G. For a real number α ∈ [ 0 , 1 ] , the generalized adjacency matrix A α (G) is defined as A α (G) = α D (G) + (1 - α) A (G) . The largest eigenvalue of the matrix A α (G) is the generalized adjacency spectral radius or the A α -spectral radius of the graph G. In this paper, we obtain some new sharp lower and upper bounds for the generalized adjacency spectral radius of G, in terms of different parameters like vertex degrees, the maximum and the second maximum degrees, the number of vertices and the number of edges, etc, associated with the structure of graph G. The extremal graphs attaining these bounds are characterized. We show that our bounds improve some recent given bounds in the literature in some cases. Further, our results extend some known results for the adjacency and/or the signless Laplacian spectral radius of a graph G to a general setting. [ABSTRACT FROM AUTHOR]

A smooth traffic flow is very crucial for an intelligent traffic system. Consequently, traffic forecasting is critical in achieving unwrinkled traffic flow. However, due to its spatial and temporal interdependence, traffic forecasting might be more difficult. Graph neural networks (GNN) are widely used to obtain traffic forecasting due to their capability to operate in non-Euclidean data. The topological connection of the traffic network plays a crucial role in graph structure learning. Therefore, the importance of the adjacency matrix in graph construction might be significant for effective traffic forecasting. As a result, for efficient graph construction, a Deep Spatial-Temporal Graph Neural Network (DSTGNN) is proposed for the construction of the weighted adjacency matrix and traffic speed prediction accurately by recording topological and temporal information. Three different weighted adjacency matrices are suggested depending on three different proposed algorithms for different variations of traffic's spatial condition. The new adjacency matrix is used in a novel DSTGNN for predicting the traffic state. The novel model comprised multiple layers with skip connections to adequately extract the variation of temporal and spatial information. DSTGNN outperforms the standard and other graph neural network models on four traffic speed datasets, SZ-taxi, Los-loop, PeMSD7, and METR-LA. [ABSTRACT FROM AUTHOR]

Let a network be represented by a simple graph G with n vertices. A common approach to investigate properties of a network is to use the adjacency matrix A = [ a ij ] i , j = 1 n ∈ R n × n associated with the graph G , where a ij > 0 if there is an edge pointing from vertex v i to vertex v j , and a ij = 0 otherwise. Both A and its positive integer powers reveal important properties of the graph. This paper proposes to study properties of a graph G by also using the path length matrix for the graph. The (i j) th entry of the path length matrix is the length of the shortest path from vertex v i to vertex v j ; if there is no path between these vertices, then the value of the entry is ∞ . Powers of the path length matrix are formed by using min-plus matrix multiplication and are important for exhibiting properties of G . We show how several known measures of communication such as closeness centrality, harmonic centrality, and eccentricity are related to the path length matrix, and we introduce new measures of communication, such as the harmonic K-centrality and global K-efficiency, where only (short) paths made up of at most K edges are taken into account. The sensitivity of the global K-efficiency to changes of the entries of the adjacency matrix also is considered. [ABSTRACT FROM AUTHOR]

The paper examines methods for assessing the distribution of elements in a stochastic matrix assuming the exponential distribution of elements in the corresponding adjacency matrix of a graph. Two cases are considered: the first assumes the homogeneity of all the graph vertices, while the second assumes the heterogeneity in the distribution of vertices with the corresponding density calculations. Hypothesis tests are formulated for the respective distributions to determine the membership of two graph vertices in the same cluster. [ABSTRACT FROM AUTHOR]

An evolutionary synthesis method to generate digital filters with low coefficient sensitivity is presented. The method uses a chromosome coding based on the graph adjacency matrix representation. It is shown that the proposed chromosome representation enables to easily verify and avoid the generation of topologically invalid and non-computable individuals during the evolutionary process. The efficiency of the proposed algorithm is tested in the synthesis of two low-pass digital filters and the results are compared with other examples found in the literature. [ABSTRACT FROM AUTHOR]

In this paper we continue a research project concerning the study of a graph from the perspective of granular computation. To be more specific, we interpret the adjacency matrix of any simple undirected graph G in terms of data information table, which is one of the most studied structures in database theory. Granular computing (abbreviated GrC) is a well-developed research field in applied and theoretical information sciences; nevertheless, in this paper we address our efforts toward a purely mathematical development of the link between GrC and graph theory. From this perspective, the well-studied notion of indiscernibility relation in GrC becomes a symmetry relation with respect to a given vertex subset in graph theory; therefore, the investigation of this symmetry relation turns out to be the main object of study in this paper. In detail, we study a simple undirected graph G by assuming a generic vertex subset W as reference system with respect to which examine the symmetry of all vertex subsets of G. The change of perspective from G without reference system to the pair ( G, W) is similar to what occurs in the transition from an affine space to a vector space. We interpret the symmetry blocks in the reference system ( G, W) as particular equivalence classes of vertices in G, and we study the geometric properties of all reference systems ( G, W), when W runs over all vertex subsets of G. We also introduce three hypergraph models and a vertex set partition lattice associated to G, by taking as general models of reference several classical notions of GrC. For all these constructions, we provide a geometric characterization and we determine their structure for basic graph families. Finally, we apply a wide part of our work to study the important case of the Petersen graph. [ABSTRACT FROM AUTHOR]

There is a type of distance-regular graph, said to be Q-polynomial. In this paper, we investigate a generalized Q-polynomial property involving a graph that is not necessarily distance-regular. We give a detailed description of an example associated with the projective geometry L N (q) . [ABSTRACT FROM AUTHOR]

The adjacency matrices of extremal 3-uniform hypergraphs occupy a significant amount of computer memory. The solution of two problems is considered: to propose an efficient way of representing and storing such matrices and to find fast algorithms that allow us to operate just with vectors of the vertex degrees and signatures (characteristics of adjacency matrices), without using adjacency matrices in memory. As part of the first task, a second-order signature that uniquely defines an extremal 3-uniform hypergraph without using its adjacency matrix is described. A mechanism for compressing the second-order signature is also proposed, which contributes to greater storage efficiency. For the second problem, a number of algorithms are presented to describe the relationship between the vector of the vertex degrees and signatures of both the first and second orders. In addition, it is shown that an arbitrary second-order signature constructed under a number of constraints always has an extremal 3-uniform hypergraph corresponding to it. [ABSTRACT FROM AUTHOR]

As an essential part of traffic management, traffic flow prediction attracts worldwide attention to intelligent traffic systems (ITSs). Complicated spatial dependencies due to the well-connected road networks and time-varying traffic dynamics make this problem extremely challenging. Recent works have focused on modeling this complicated spatial-temporal dependence through graph neural networks with a fixed weighted graph or an adaptive adjacency matrix. However, fixed graph methods cannot address data drift due to changes in the road network structure, and adaptive methods are time consuming and prone to be overfitting because the learning algorithm thoroughly optimizes the adaptive matrix. To address this issue, we propose a principal graph embedding convolutional recurrent network (PGECRN) for accurate traffic flow prediction. First, we propose the adjacency matrix graph embedding (AMGE) generation algorithm to solve the data drift problem. AMGE can model the distribution of spatiotemporal series after data drift by extracting the principal components of the original adjacency matrix and performing an adaptive transformation. At the same time, it has fewer parameters, alleviating overfitting. Then, except for the essential spatial correlations, traffic flow data are also temporally dynamic. We utilize temporal variation by integrating gated recurrent units (GRU) and AMGE to comprise the proposed model. Finally, PGECRN is evaluated on two real-world highway datasets, PeMSD4 and PeMSD8. Compared with the existing baselines, the better prediction accuracy of our model shows that it can accurately and efficiently model traffic flow. [ABSTRACT FROM AUTHOR]

An additive scheme with 11 constants is derived from the coefficients of characteristic polynomials (CCPs) of adjacency matrix A of irregular molecular graphs (IMG) for molecules that contain a bivalent heteroatom -SH or -OH at the beginning of the chain. The structural significance of the CCPs to adjacency matrix A′ is established. Our formula is used to calculate the enthalpies of formation Δ H of liquid alkanethiols (mercaptanes) CHSH and Δ H of liquid saturated monoalcohols CHOH that remain unstudied experimentally. [ABSTRACT FROM AUTHOR]

This paper presents a system for automatic generation of the adjacency matrix from the image of graphs. The graph, we assume, is printed or hand printed and available as a part of a document either separately or along with text and picture. A morphology-based approach is used here to separate components of the graphs: vertices, edges and labels. A novel technique is proposed to traverse the nonplanar edges joining the vertices. The proposed method may be used for logical compression of the information contained in the graph image in the form of an adjacency matrix. It may also be used to replace the cumbersome, error-prone and time-consuming manual method of generation of the adjacency matrix for graphs with large number of vertices and complex interconnections. [ABSTRACT FROM AUTHOR]