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Let G ˆ be a mixed graph and α ∈ [ 0 , 1 ]. Let D ˆ (G ˆ) and A ˆ (G ˆ) be the diagonal matrix of vertex degrees and the mixed adjacency matrix of G ˆ , respectively. The α -mixed adjacency matrix of G ˆ is the matrix A ˆ α (G ˆ) = α D ˆ (G ˆ) + (1 − α) A ˆ (G ˆ). We study some properties of A ˆ α (G ˆ) associated with some type of mixed graphs, namely quasi-bipartite and pre-bipartite mixed graphs. A spectral characterization for pre-bipartite and some class of quasi-bipartite mixed graphs is given. For a mixed graph G ˆ we exploit the problem of finding the smallest α for which A ˆ α (G ˆ) is positive semi-definite. This problem was proposed by Nikiforov in the context of undirected graphs. It is proven here that, for a mixed graph this number is not greater than 1 2 and that a connected mixed graph G ˆ with n ≥ 2 is quasi-bipartite if and only if this number is exactly 1 2. The spread of the α -mixed adjacency matrix is the difference among the largest and the smallest α -mixed adjacency eigenvalue. Upper and lower bounds for the spread of the α - mixed adjacency matrix are obtained. The α -mixed Estrada index of G ˆ is the sum of the exponentials of the eigenvalues of A ˆ α (G ˆ). In this paper, bounds for the eigenvalues of A ˆ α (G ˆ) are established and, using these bounds some sharp bounds on the mixed Estrada index of A ˆ α (G ˆ) are presented. [ABSTRACT FROM AUTHOR]

Let f (x , y) > 0 be a real symmetric function. For a connected graph G , the weight of edge v i v j is equal to the value f (d i , d j) , where d i is the degree of vertex v i. The degree-based weighted adjacency matrix is defined as A f (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 0 otherwise. In this paper, we first give some bounds of the weighted adjacency eigenvalue λ 1 (A f (G)) in terms of λ 1 (A f (H)) , where H is obtained from G by some kinds of graph operations, including deleting vertices, deleting an edge and subdividing an edge, and examples are given to show that bounds are tight. Second, we obtain some bounds for the largest weighted adjacency eigenvalue λ 1 (A f (G)) of irregular weighted graphs. [ABSTRACT FROM AUTHOR]

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

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]

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]

Different from other computer vision tasks, action recognition needs to process larger-scale video data. How to extract and analyze the effective parts from a huge amount of video information is the main difficulty of action recognition technology. In recent years, due to the outstanding performance of Graph Convolutional Networks (GCN) in many fields, a new solution to the action recognition algorithm has emerged. However, in current GCN models, the constant physical adjacency matrix makes it difficult to mine synergistic relationships between key points that are not directly connected in physical space. Additionally, a simple time connection of skeleton data from different frames makes each frame in the video contribute equally to the recognition results, which increases the difficulty of distinguishing action stages. In this paper, the information extraction ability of the model has been optimized in the space domain and time domain, respectively. In the space domain, an Adjacency Matrix Generation (AMG) module, which can pre-analyze node sets and generate an adaptive adjacency matrix, has been proposed. The adaptive adjacency matrix can help the graph convolution model to extract the synergistic information between the key points that are crucial for recognition. In the time domain, the Time Domain Attention (TDA) mechanism has been designed to calculate the time-domain weight vector through double pooling channels and complete the weights of key point sequences. Furthermore, performance of the improved TDA-AMG-GCN modules has been verified on the NTU-RGB+D dataset. Its detection accuracy at the CS and CV divisions reached 84.5% and 89.8%, respectively, with an average level higher than other commonly used detection methods at present. [ABSTRACT FROM AUTHOR]

With the deterioration growth of water pipes, the frequency of identifying segments and unintended isolation in water distribution networks is expected to increase, because it can help identify the number of nodes that cannot be supplied and their locations when a failure occurs. Therefore, a methodology for identifying segments and unintended isolation was developed and verified using examples. In addition, a case study was conducted to confirm that the proposed methodology could be applied to actual fields. The proposed method only requires the node-based adjacency matrix K for identification, where the connectivity of the current network can be intuitively understood with the constructed matrices. Moreover, this methodology is easy to adopt using the single row-first search method. The proposed framework is compatible with the data storage format of hydraulic analysis software. Finally, the proposed methodology can be used in various research on configuring and analyzing water distribution networks. [ABSTRACT FROM AUTHOR]

Graph convolutional networks (GCNs), which extend convolutional neural networks (CNNs) to non-Euclidean structures, have been utilized to promote skeleton-based human action recognition research and have made substantial progress in doing so. However, there are still some challenges in the construction of recognition models based on GCNs. In this paper, we propose an enhanced adjacency matrix-based graph convolutional network with a combinatorial attention mechanism (CA-EAMGCN) for skeleton-based action recognition. Firstly, an enhanced adjacency matrix is constructed to expand the model's perceptive field of global node features. Secondly, a feature selection fusion module (FSFM) is designed to provide an optimal fusion ratio for multiple input features of the model. Finally, a combinatorial attention mechanism is devised. Specifically, our spatial-temporal (ST) attention module and limb attention module (LAM) are integrated into a multi-input branch and a mainstream network of the proposed model, respectively. Extensive experiments on three large-scale datasets, namely the NTU RGB+D 60, NTU RGB+D 120 and UAV-Human datasets, show that the proposed model takes into account both requirements of light weight and recognition accuracy. This demonstrates the effectiveness of our method. [ABSTRACT FROM AUTHOR]

The extended adjacency matrix of a graph has been introduced to expand some beneficial molecular topological indices. In the present work, the spectra of certain classes of vertex or edge-transitive graphs are investigated. Also, we study some spectral properties of extended adjacency matrix. Besides, we study the spectral A e x -spread of a graph. In this way, we explore some new bounds for both the smallest and the largest eigenvalues. Finally, we persue the behaviour of A e x -energy of a graph. [ABSTRACT FROM AUTHOR]

Let D be a strongly connected digraph and α ∈ [ 0 , 1 ]. In Liu et al. (2019) [13] the matrix A α (D) = α D e g (D) + (1 − α) A (D) , where A (D) and D e g (D) are the adjacency matrix and the diagonal matrix of the out-degrees of D , respectively, was defined. In this paper it is established some sharp bounds on the A α (D) -spectral radius in terms of some parameters such as the out-degrees, the maximum out-degree, the second maximum out-degree, the number of vertices, the number of arcs, the average 2-outdegrees of the vertices of D and the parameter α of A α (D). The extremal digraphs attaining these bounds are characterized. It is shown that the bounds obtained improve, in some cases, some of recently given bounds presented in the literature. [ABSTRACT FROM AUTHOR]

Accurate forecasting of taxi demand has facilitated the rational allocation of urban public transport resources, reduced congestion in urban transport networks, and shortened passenger waiting time. However, virtual station discovery and modelling of the demand when forecasting through graph convolutional neural networks remains challenging. In this study, the virtual station discovery problem was addressed by using a two-stage clustering approach, which considers the geographical and load characteristics of taxi demand. Furthermore, a fusion model combining non-negative matrix decomposition and a graph convolutional neural network was proposed in order to extract the features of the nodes for dimension reduction and adaptive adjacency matrix computation. By the construction of a local processing structure, further extraction of the local characteristics of the demand was achieved. The experimental results show that the method in this study outperforms state-of-the-art methods in terms of the root mean square error and average absolute value error. Therefore, the model proposed in this study is able to achieve accurate forecasting of taxi demand. [ABSTRACT FROM AUTHOR]

Modeling complex spatial and temporal dependencies in multivariate time series data is crucial for traffic forecasting. Graph convolutional networks have proved to be effective in predicting multivariate time series. Although a predefined graph structure can help the model converge to good results quickly, it also limits the further improvement of the model due to its stationary state. In addition, current methods may not converge on some datasets due to the graph structure of these datasets being difficult to learn. Motivated by this, we propose a novel model named Dynamic Correlation Graph Convolutional Network (DCGCN) in this paper. The model can construct adjacency matrices from input data using a correlation coefficient; thus, dynamic correlation graph convolution is used for capturing spatial dependencies. Meanwhile, gated temporal convolution is used for modeling temporal dependencies. Finally, we performed extensive experiments to evaluate the performance of our proposed method against ten existing well-recognized baseline methods using two original and four public datasets. [ABSTRACT FROM AUTHOR]

Let A (G) be the adjacency matrix of a graph G. Let s (v) denote the row entries of A (G) corresponding to the vertex v of G. The Hamming distance between the strings s (u) and s (v) is the number of positions in which s (u) and s (v) differ. In this paper, we study the Hamming distance between the strings generated by the adjacency matrix of subgraph complement of a graph. We also compute sum of Hamming distances between all pairs of strings generated by the adjacency matrix of G ⊕ S. [ABSTRACT FROM AUTHOR]

The research on node importance identification for temporal networks has attracted much attention. In this work, combined with the multi-layer coupled network analysis method, an optimized supra-adjacency matrix (OSAM) modeling method was proposed. In the process of constructing an optimized super adjacency matrix, the intra-layer relationship matrixes were improved by introducing the edge weight. The inter-layer relationship matrixes were formed by improved similarly and the inter-layer relationship is directional by using the characteristics of directed graphs. The model established by the OSAM method accurately expresses the structure of the temporal network and considers the influence of intra- and inter-layer relationships on the importance of nodes. In addition, an index was calculated by the average of the sum of the eigenvector centrality indices for a node in each layer and the node importance sorted list was obtained from this index to express the global importance of nodes in temporal networks. The experimental results on three real temporal network datasets Enron, Emaildept3, and Workspace showed that compared with the SAM and the SSAM methods, the OSAM method has a faster message propagation rate and larger message coverage and better SIR and NDCG@10 indicators. [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]

In monocular 3D human pose estimation, modeling the temporal relation of human joints is crucial for prediction accuracy. Currently, most methods utilize transformer to model the temporal relation among joints. However, existing transformer-based methods have limitations. The temporal adjacency matrix utilized within the self-attention of the temporal transformer inaccurately models the temporal relationships between frames, particularly in cases where distinct motions exhibit significant correlation despite having different physical interpretations and large temporal spans. To address this issue, we construct an artificial temporal adjacency matrix based on input data and introduce a temporal adjacency matrix hybrid module to blend this matrix with the model's inherent temporal adjacency matrix, resulting in a novel composite temporal adjacency matrix. Through extensive experiments on Human3.6M and MPI-INF-3DHP datasets using state-of-the-art methods as benchmarks, our proposed method demonstrates a maximum improvement of up to 5.6% compared to the original approach. [ABSTRACT FROM AUTHOR]

An extend adjacency matrix of a graph ( A e x ) was introduced decades ago as a precursor for developing a few quite useful molecular topological descriptors. The spectral radius ( η 1 ) of the extended adjacency matrix and the extended energy of a graph ( E e x ) have been successfully utilized in QSPR/QSAR investigations. Here, the η 1 and E e x have been further mathematically analyzed. Several sharp upper bounds for the E e x are obtained. In addition, the Nordhaus–Gaddum-type results for the η 1 and E e x are presented. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n. The matrix S(G) = D(G) + A(G) is called the signless Laplacian matrix of G, where D(G) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let s1(G) and λ1(G) be the largest eigenvalue of S(G) and A(G), respectively. In this paper, we first present sharp upper and lower bounds for s1(G) and λ1(G) involving the maximum degree, the minimum degree, order, size and sum-connectivity F-index. Moreover, we investigate the relation between s1(G) and λ1(G). [ABSTRACT FROM AUTHOR]

Recent progress on human action recognition, fueled by the Graph Convolutional Network (GCN), has been substantial. However, two main problems are caused by the design strategy of graph convolution kernels: first, the partitioning strategy of neighbor set for graph vertices relies on the gravity center designed manually, which is limited in generalizability to diverse skeletons in action recognition; second, the existing GCN-based methods can only capture local physical dependencies among joints and result in missing implicit joint correlations due to over-smoothing. In this work, we present (1) a novel attention adjacency matrix (AAM) to design graph convolution kernels and (2) a dimension-attention block to improve the robustness of the model. Specifically, the proposed AAM is designed by a novel partitioning strategy for the neighbor set, through which an adjacency matrix is decomposed into several parametric matrices. Simultaneously, attention mechanism is introduced in the process to generate an attention matrix. Combining the matrix and the parametric matrices into an AAM through ResNet, we further exhibit the AAM based graph convolution network (AAM-GCN). The proposed dimension-attention block strengthens the important information in each dimension of skeleton data by extending the idea of channel-attention. Extensive experiments on two large-scale datasets, NTU-RGB+D and Kinetics, demonstrate that AAM-GCN achieves better performance than the state-of-the-art works. [ABSTRACT FROM AUTHOR]

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of a complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron–Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of a gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same. [ABSTRACT FROM AUTHOR]

The extended adjacency matrix Ae (G) of a graph G is determined by graph degrees. Some inequalities are found for the extended adjacency matrix including its vertices, its edges and its degrees in this paper. Also, some bounds are established for this matrix involving its energy and its laplacian energy. [ABSTRACT FROM AUTHOR]

Fast and accurate ellipse detection is critical in certain computer vision tasks. In this paper, we propose an arc adjacency matrix-based ellipse detection (AAMED) method to fulfill this requirement. At first, after segmenting the edges into elliptic arcs, the digraph-based arc adjacency matrix (AAM) is constructed to describe their triple sequential adjacency states. Curvature and region constraints are employed to make the AAM sparse. Secondly, through bidirectionally searching the AAM, we can get all arc combinations which are probably true ellipse candidates. The cumulative-factor (CF) based cumulative matrices (CM) are worked out simultaneously. CF is irrelative to the image context and can be pre-calculated. CM is related to the arcs or arc combinations and can be calculated by the addition or subtraction of CF. Then the ellipses are efficiently fitted from these candidates through twice eigendecomposition of CM using Jacobi method. Finally, a comprehensive validation score is proposed to eliminate false ellipses effectively. The score is mainly influenced by the constraints about adaptive shape, tangent similarity, distribution compensation. Experiments show that our method outperforms the 12 state-of-the-art methods on 9 datasets as a whole, with reference to recall, precision, F-measure, and time-consumption. [ABSTRACT FROM AUTHOR]

Accurate and real-time traffic forecasting is of great significance for urban traffic planning, traffic control, and traffic management. However, the time-varying dynamic spatial relations and the complicated spatial–temporal dependencies are still open problems to be considered in traffic forecasting. To address these issues, we propose a graph convolutional recurrent network for traffic forecasting with a dynamic adjacency matrix within an encoder–decoder framework, named DAGCRN. The DAGCRN consists of a spatial relation extraction module (SREM), an adjacency matrix update module (AMUM), a dynamic graph convolutional recurrent module (DGCRM), and a global temporal attention module (GTAM). Specifically, SREM and AMUM are proposed to capture nodes' mutual relations at each time step and to model the evolution of the dynamic adjacency matrix, respectively. DGCRM captures the spatial–temporal dependencies of traffic data based on dynamic graph convolution and gated recurrent unit. GTAM is designed to extract the long-range temporal dependencies between future time steps and historical time steps. Extensive experiments on two real-world traffic speed datasets demonstrate that the proposed DAGCRN outperforms a number of representative baselines consistently. [ABSTRACT FROM AUTHOR]

Minimal codewords have applications in decoding linear codes and in cryptography. We study the number of minimal codewords in binary linear codes that arise by appending a unit matrix to the adjacency matrix of a graph. [ABSTRACT FROM AUTHOR]

The utilization of extended adjacency matrices to investigate various combinatorial properties of graph is one of the most recent research directions in the spectral graph theory. This work proposes a useful modification to the classical adjacency matrix, as well as associated spectral radius and energy. When compared to existing descriptors, the novel energy and Estrada index are established to be significant molecular descriptors in structure–property modelling and isomer discrimination by means of quantitative structure property relationship (QSPR) analysis. Following that, mathematical attributes of the energy are demonstrated by estimating its tight bounds with identifying extremal graphs. • The neighbourhood arithmetic-geometric energy ( E NAG) is proposed. • The chemical significance of E NAG is studied. • It outperforms the existing descriptors in isomer discrimination. • Mathematical properties of this invariant is investigated. • Estrada index of E NAG is also investigated. [ABSTRACT FROM AUTHOR]

We obtain bounds for the largest and least eigenvalues of the adjacency matrix of a simple undirected graph. We find upper bound for the second largest eigenvalue of the adjacency matrix. We prove that the bounds obtained here improve on the existing bounds and also illustrate them with examples. [ABSTRACT FROM AUTHOR]

Abstract In this paper we introduce a new graph matrix, named the anti-adjacency matrix or eccentricity matrix, which is constructed from the distance matrix of a graph by keeping for each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping for each row and each column only the distances equal to 1. We show that the eccentricity matrix of trees is irreducible, and we investigate the relations between the eigenvalues of the adjacency and eccentricity matrices. Finally, we give some applications of this new matrix in terms of molecular descriptors, and we conclude by proposing some further research problems. [ABSTRACT FROM AUTHOR]

The use of complex networks as a modern approach to understanding the world and its dynamics is well-established in the literature. The adjacency matrix, which provides a one-to-one representation of a complex network, can also yield several metrics of the graph. However, it is not always clear whether this representation is unique, as the permutation of rows and columns in the matrix can represent the same graph. To address this issue, the proposed methodology employs a sorting algorithm to rearrange the elements of the adjacency matrix of a complex graph in a specific order. The resulting sorted adjacency matrix is then used as input for feature extraction and machine learning algorithms to classify the networks. The results indicate that the proposed methodology outperforms previous literature results on synthetic and real-world data. • This work presents a new technique for extracting important features of complex networks to improve machine learning performance. • The proposed methodology employs sorting algorithm to rearrange the elements of adjacency matrix of complex graph. • The resulting sorted adjacency matrix is then used as input for feature extraction and machine learning classification. • Results indicate that the proposed methodology outperforms previous literature results in synthetic and real-world data. [ABSTRACT FROM AUTHOR]

In water distribution networks (WDNs), pipe failure is the most frequent type of equipment malfunction, resulting in water supply suspension. To determine the water suspension area and to properly plan restoration work, the accurate identification of isolated subsystems with respect to valve locations is critical. The water suspension area can be divided into intended and unintended isolations, depending on the valve layout. This technical note discusses the conventional method of searching the valve-controlled isolation areas and specifically examines the weakness of the unintended isolation (UI) search algorithm that uses a node-by-node adjacency matrix. In this study, we propose a new search algorithm using a link-by-link adjacency matrix to identify the UI segments. Demonstrations on branch and loop networks show that the new approach correctly identifies the UI in various operating conditions. The proposed method is expected to replace the conventional method and can be applied in various valve-controlled segment analyses, such as optimal valve placement, real-time valve control for dynamic WDN operation, and postaccident pipe recovery planning. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n , without isolated vertices. Denote by A = (a i j) n × n the adjacency matrix of G. Eigenvalues of the matrix A , λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n , form the spectrum of the graph G. An important spectrum-based invariant is the graph energy, defined as E (G) = ∑ i = 1 n | λ i |. The determinant of the matrix A can be calculated as det A = ∏ i = 1 n λ i . Recently, Altindag and Bozkurt [Lower bounds for the energy of (bipartite) graphs, MATCH Commun. Math. Comput. Chem.77 (2017) 9–14] improved some well-known bounds on the graph energy. In this paper, several inequalities involving the graph invariants E (G) and | det A | are derived. Consequently, all the bounds established in the aforementioned paper are improved. [ABSTRACT FROM AUTHOR]

Abstract Let G be a graph of order n and let d i be the degree of the vertex v i in G for i = 1 , 2 , … , n. The weighted adjacency matrix A db of G is defined so that its (i, j)-entry is equal to d i + d j d i d j if the vertices v i and v j are adjacent, and 0 otherwise. The spectral radius ϱ 1 and the energy E d b of the A db -matrix are examined. Lower and upper bounds on ϱ 1 and E d b are obtained, and the respective extremal graphs are characterized. [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]

The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. If D is a digraph of order n with eigenvalues z1, z2, ... , zn, then its energy is defined by E(D) = Σnk =1 ∣ Re(zk ) ∣ . We give a new notion of energy of digraphs defined by Ec(D) = Σnk =1 ∣ Im(zk ) ∣ and call it the iota energy of the digraph D. It is shown that the Coulson's integral formula remains valid for iota energy. We also find the unicyclic digraphs with extremal iota energies among the class of unicyclic digraphs with a fixed order. Furthermore, it is shown that the iota energy is increasing over the set Dn,h of n-vertex digraphs with cycles of length h, with respect to a quasi-order relation. We also generalize the increasing property of the energy over the set Dn,h with respect to quasi-order relation. [ABSTRACT FROM AUTHOR]

Molecular network plays a critical role in determining dynamic elasticity of soft condensed polymers, e.g., their varied cross-linking densities and end-to-end distances with changed Young's moduli. Although it has been studied for decades, the coupling relationship between vertices and edges of molecular networks is not well understood, mainly because the degree of the crosslinking points and asymmetry molecular networks have not been considered in the previously models. In this study, a graph theory was employed to formulate an undirected graphical model and describe the coupling between vertices and edges in molecular networks, based on which the dynamic elasticity of polyelectrolyte hydrogel was modeled. From the Kirchhoff graph theory and bead-spring model, the coupling relationship between vertices and edges was obtained using end-to-end distance and viscosity parameters. Combining the Watts–Strogatz model and kinetic probability, the coupling between vertices and edges for the polyelectrolyte network was studied. Furthermore, an adjacency matrix with eigenvalue, number of vertices and mean degree was proposed to formulate constitutive relationships including dynamic elasticity and stress-strain, according to rubber elasticity theory and Mooney-Rivlin model, respectively. The linking between the vertices and edges determines the network structure and dynamic elasticity of the polyelectrolyte hydrogel. Based on the graph theory, the vertices and edges are encoded by adjacency matrix, which is proposed to describe the dynamic elasticity of symmetric and asymmetric network structures using the crosslinking density and end-to-end distance. Finally, effectiveness of the undirected graphical model was verified using both finite element analysis and experimental results of polyelectrolyte hydrogels reported in literature. Constitutive stress-elongation ratio curves for the polyelectrolyte hydrogel, of which the graph network is determined by the mean degree of d = 2, d = 3, d = 4, d = 5, d = 6 and d = 7, at the same vertex number of N = 8 in molecular network. [Display omitted] • An undirected graphical model and describe the coupling between vertices and edges in molecular networks. • Dynamic elasticity of polyelectrolyte hydrogel has been modeled using the adjacency matrix. • A constitutive relationship is developed to understand the molecular networks and their dynamic elasticities. [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]

This work develops properties of the recurrence sequences defined by using the characteristic polynomial of the adjacency matrix of the Cayley diagram of the dihedral group. The study of this sequence modulo m yields cyclic groups and semigroups from generating matrix. Also, we extend the sequences defined to groups and then we examine these sequences in finite groups. Finally, we obtain the periods of the extended sequences in the dihedral group D2n as applications of the results obtained. [ABSTRACT FROM AUTHOR]

In a deregulated electric power system, it is important to distribute transmission losses between market participants fairly. This paper presents an effective method for active loss allocation by tracing power flow. In this method, after obtaining the alternating current (AC) load flow results of the system, it is first converted to a system in which the power flow is constant along each branch. A new matrix (i.e., power adjacency matrix) is then constructed for the system. The matrix is decoupled, and matrix operations are carried out using the submatrices to derive three other matrices. The contributions of the generators and loads of the main system to active losses of its branches and the shares of the generators in active power consumption of the loads are specified using these matrices. The proposed method is implemented on a four-bus system and several standard test systems. The results show that the method can be applied to any transmission system, and the loss allocation results are preferable to that of previous methods. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n . For i = 1 , 2 , … , n , let d i be the degree of the vertex v i of G . The extended adjacency matrix A ex of G is defined so that its ( i, j )-entry is equal to 1 2 ( d i d j + d j d i ) if the vertices v i and v j are adjacent, and 0 otherwise,Yang et al. (1994). The spectral radius η 1 and the energy E e x of the A ex -matrix are examined. Lower and upper bounds on η 1 and E e x are obtained, and the respective extremal graphs characterized. [ABSTRACT FROM AUTHOR]

The connectivity of complex networks is usually determined by a small fraction of key nodes. Earlier works successfully identify an influential single node, yet have some problems for the case of multiple ones. In this paper, based on the matrix spectral theory, we propose the collective influence of multiple nodes. An interesting finding is that some traditionally influential nodes have strong internal coupling interactions that reduce their collective influence. We then propose a greedy algorithm to dismantle complex networks by optimizing the collective influence of multiple nodes. Experimental results show that our proposed method outperforms the state of the art methods in terms of the principal eigenvalue and the giant component of the remaining networks. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V(G) = {v1,v2,..., vn}. Let A(G) be the adjacency matrix of a graph G. The rows of A(G) corresponding to a vertex v of G, denoted by s(v) is the string. The Hamming index of a graph G is the sum of the Hamming distances between all pairs of vertices of G. In this paper we obtain Hamming index generated by adjacency matrix of some thorn graphs. [ABSTRACT FROM AUTHOR]

A mixed graph is a graph with edges and arcs, which can be considered as a hybrid of an undirected graph and a directed graph. In this paper we define the mixed adjacency matrix and the mixed energy of a mixed graph. The mixed adjacency matrix generalizes both the adjacency matrix of an undirected graph and the skew-adjacency matrix of a digraph. Then we compute the characteristic polynomial of the mixed adjacency matrix of a mixed graph and deduce some basic results from it. Furthermore, we give bounds to the mixed energy of a general mixed graph, and we compute the mixed energy of some special mixed graphs. At the end of the paper, we introduce mixed unitary Cayley graphs and compute their spectra. [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]

Let Gσ be an oriented graph with underlying simple graph G. The skew-adjacency matrix of Gσ is the {0, 1,-1}-matrix S = S(Gσ) = [sij ], such that sij = 1 if (vi, vj) is an arc in Gσ, sij = -1 if (vj, vi) is an arc in Gσ and sij = 0, otherwise. In this paper, all connected oriented graphs with three distinct skew-eigenvalues 0 and ±2i are characterized. [ABSTRACT FROM AUTHOR]

The demolition issue of long-span spatial lattice structures has become increasingly prominent with the rapid urban development and building upgrades. Building deconstruction offers superior economic and environmental benefits compared to traditional demolition and therefore should be further promoted and applied. Dismantling in blocks is an efficient approach for deconstruction of long-span spatial lattice structures, surpassing the practice of dismantling individual components one by one. However, dividing the entire structure into substructures considering the convenience and safety of dismantling operations is the first challenge of dismantling in blocks yet to be overcome. In this study, a block planning method was proposed to aid practitioners in making high-quality block schemes. The topological relation of spatial grids defined by the adjacency matrix is the basis. The comprehensive evaluation index was established reflecting the operability, efficiency, and safety of the block scheme. Taking the combination of spatial grids as the variable and the comprehensive evaluation index combined with constraint functions as the objective, the block planning was formulated as an optimization problem. The genetic algorithm was adopted to solve this problem, and two examples were presented for verification. Results confirmed that the proposed block planning method is effective to generate the high-quality block scheme that meets topology, quantity, and lifting weight requirements and has great synthetical performance. The consideration of structural symmetry can accelerate the optimization process. The zoning planning strategy provides a tool for practitioners to make the block scheme that meets the lifting weight requirements of different zones when the crane's lifting capacity is limited by operating space. The proposed block planning method will enrich the construction management technologies of dismantling projects and help engineers make efficient and wise decisions. [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]

Although eigenvectors belong to the core of linear algebra, relatively few closed-form expressions exist, which we bundle and discuss here. A particular goal is their interpretation for graph-related matrices, such as the adjacency matrix of an undirected, possibly weighted graph. [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]