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All graphs under consideration are finite, simple, connected, and undirected. Adjacency matrix of a graph G is 0,1 matrix A = a i j = 0 , i f v i = v j o r d v i , v j ≥ 2 1 , i f d v i , v j = 1. . Here in this paper, we discussed new type of adjacency matrix known by 1-2 adjacency matrix defined as A 1,2 G = a i j = 0 , i f v i = v j o r d v i , v j ≥ 3 1 , i f d v i , v j = 2 , from eigenvalues of the graph, we mean eigenvalues of the 1-2 adjacency matrix. Let T n c be the set of the complement of trees of order n . In this paper, we characterized a unique graph whose least eigenvalue is minimal among all the graphs in T n c .

Let $ D$ be a strongly connected digraph and $\alpha\in [0,1].$ In [J. P. Liu, X. Z. Wu, J. S. Chen and B. L. Liu, The $ A_{\alpha} $ spectral radius characterization of some digraphs, Linear Algebra Appl. 563 (2019) 63--74] the matrix $ A_{\alpha}(D)=\alpha Deg(D)+(1-\alpha)A(D),$ where $ A(D)$ and $Deg(D)$ 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 $A_{\alpha}(D)$-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 $ \alpha$ of $A_{\alpha}(D)$. 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. in publication

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.

Recently, discovering helpful information from a database consisting of transactions has been a critical research topic. Several frequent itemsets mining for association rule mining, algorithms that can only handle binary databases have been proposed. Transactions using numerical values, on the other hand, are ubiquitous in real-world applications. Thus, with reference to the quantitative transactional database, several algorithms were developed and “fuzzy frequent itemsets” (FFI) were discovered. Most of them just consider the term having maximum cardinality. As a result, the number of fuzzy regions processed is equal to the number of original elements. Multiple fuzzy zones of an item, on the other hand, give a better result for making a correct decision. This study presents an AMFFI-miner (Adjacency matrix-based multiple fuzzy frequent itemsets) for discovering multiple FFIs out of a quantitative transactional database. Adjacency matrix and fuzzy-list structures were designed to find multiple FFIs by scanning database only ones and lesser candidate itemsets generation. Join two nodes if its co-occurrence between two fuzzy linguistics terms satisfies minsupport threshold by finding the co-occurrence between two fuzzy linguistics terms directly from the adjacency matrix, thus reducing the number of node joining and speeding up discovering multiple FFI. Experiments were carried out to compare the suggested method's performance to that of existing methodologies on the basis of running time, memory utilization, and the number of nodes joining.

The traditional method of searching transmission section generally has the problem of omission and time-consuming. A topology converging algorithm of power grid adjacency matrix is proposed in this paper. It aggregates neighboring buses based on a simple matrix transformation rule, and identify key transmission sections based on matrix operations and power flow distribution factors. This method has the advantage of no presetting the sub-zone, which can avoid the simplification of the grid topology in the sub-zone and improve the calculation accuracy. The complex logical operations of the traversal algorithm adopted by most of traditional methods can be avoided, and the calculation speed can be effectively improved. Besides, the algorithm also has a memory characteristic which can be applied to the online search of key transmission sections. The results of several IEEE test systems show the effectiveness of the proposed algorithm.

In this article, proposed a algorithm to solve the membership problem in Hyperedge Replacement Grammars (HRG). Given a hypergraph H with labeled nodes rooted and directed hyperedges, the problem consists in determining if H 2 L(G), where G is in HRG, this is to say, if H is in the language generated by G, for this the analysis is done directly in the Adjacency Matrix of the hypergraph H. For the algorithm proposed, also presents the proof of its correctness.

This paper presents a problem of community detection in sparse network. Graph represents the network with {0, 1} symmetric matrix, this matrix is defined to be sparse when most of its entries are zeros. The problem of community detection of this type of networks is non-deterministic polynomial-time hardness (NP-hard) problem. Here, we give a simple idea to regularize the sparse matrix by adding a heuristic parameter to the entries of the matrix. This work performs integrating Tabu Search via Fuzzy C-mean to compute variants of the modularity maximization. The results show the ability of the proposed method to define structure of the network by optimizing different types of the quality functions; the results show the global function gives the high value in must runs when apply it on a large sparse real networks.

The paper presents a novel spectral algorithm EVSA (eigenvector structure analysis), which uses eigenvalues and eigenvectors of the adjacency matrix in order to discover clusters. Based on matrix perturbation theory and properties of graph spectra we show that the adjacency matrix can be more suitable for partitioning than other Laplacian matrices. The main problem concerning the use of the adjacency matrix is the selection of the appropriate eigenvectors. We thus propose an approach based on analysis of the adjacency matrix spectrum and eigenvector pairwise correlations. Formulated rules and heuristics allow choosing the right eigenvectors representing clusters, i.e., automatically establishing the number of groups. The algorithm requires only one parameter-the number of nearest neighbors. Unlike many other spectral methods, our solution does not need an additional clustering algorithm for final partitioning. We evaluate the proposed approach using real-world datasets of different sizes. Its performance is competitive to other both standard and new solutions, which require the number of clusters to be given as an input parameter.

In this paper, the consensus problem for the first-order discrete-time multi-agent systems (MASs) In the presence of fault is studied. This study uses two concepts of watchdog model-based observer and weighted adjacency matrix approach (WAMA). The purpose of this study is to reach a consensus on MASs when one or more agents have been damaged. In this method, for any agent, a watchdog that measures the amount of difference between the states of the actual system and the calculated states of the system by the agent's model is designed, this generated residual declares the amount of agents deviation. A decision function is used to decide how this malfunctioning affects the neighboring agents. Therefore, the states that each agent transmits to the neighboring agents are augmented by the output of the decision function. By increasing states matrix dimensions, each agent will report its health status or commanding by using a new state to the neighboring agents. This additional state is affected by a decision function that expresses the neighboring agents how should behave with this malfunction agent. Finally, the produced residue will be weight of the adjacency matrix and will cause to correct consensus. Simulation results are presented to illustrate the effectiveness of this approach.

Spectral graph theory uses the eigenvalues of matrices associated with a graph to determine the structural properties of the graph. The spectrum of the generalized adjacency matrix is considered in the paper. Graphs with the same spectrum are called cospectral. Is every graph uniquely determined by its spectrum (DS for short)? This question goes back for about half a century, and originates from chemistry. In 1956 Gunthard and Primas raised the question in a paper that related the theory of graph spectra to Huckel’s theory. At that time it was believed that every graph is determined by the spectrum, until in 1957 Collatz and Sinogowitz presented a pair of cospectral trees. In 1967 Schwenk proved that for almost all trees there is another tree with the same spectrum. Such a statement is neither proved nor refuted for the class of graphs in general. Till now, computational experiments were done on the set of all graphs on up to 12 vertices by Haemers. Computer enumerations for small n show that up to 10 vertices the fraction of graphs that are DS decreases, but for n = 11 and n = 12 it increases again. We consider the construction of the cospectral graphs called GM-switching for graph G taking the cycle C2n and adjoining a vertex v adjacent to half the vertices of C2n. For these graphs we determine the pairs of cospectral nonisomorphic graphs for small n. It is an operation on graphs that leaves the spectrum of the generalized adjacency matrix invariant. It turns out that for the enumerated cases a large part of all cospectral graphs comes from GM switching, and that the fraction of graphs on n vertices with a cospectral mate starts to decrease at some value of n < 11 (depending on the matrix). Since the fraction of cospectral graphs on n vertices constructible by GM switching tends to 0 if n → ∞, the present data give some indication that possibly almost no graph has a cospectral mate. Haemers and Spence derived asymptotic lower bounds for the number of graphs with a cospectral mate from GM switching.

This work revisits the information given by the graph-of-words and its typical utilization through graph-based ranking approaches in the context of keyword extraction. Recent, well-known graph-based approaches typically employ the knowledge from word vector representations during the ranking process via popular centrality measures (e.g., PageRank) without giving the primary role to vectors’ distribution. We consider the adjacency matrix that corresponds to the graph-of-words of a target text document as the vector representation of its vocabulary. We propose the distribution-based modeling of this adjacency matrix using unsupervised (learning) algorithms. The efficacy of the distribution-based modeling approaches compared to state-of-the-art graph-based methods is confirmed by an extensive experimental study according to the F1 score. Our code is available on GitHub.

Background: Post-translational modification (PTM) is a biological process that is associated with the modification of proteome, which results in the alteration of normal cell biology and pathogenesis. There have been numerous PTM reports in recent years, out of which, lysine phosphoglycerylation has emerged as one of the recent developments. The traditional methods of identifying phosphoglycerylated residues, which are experimental procedures such as mass spectrometry, have shown to be time-consuming and cost-inefficient, despite the abundance of proteins being sequenced in this post-genomic era. Due to these drawbacks, computational techniques are being sought to establish an effective identification system of phosphoglycerylated lysine residues. The development of a predictor for phosphoglycerylation prediction is not a first, but it is necessary as the latest predictor falls short in adequately detecting phosphoglycerylated and non-phosphoglycerylated lysine residues. Results: In this work, we introduce a new predictor named RAM-PGK, which uses sequence-based information relating to amino acid residues to predict phosphoglycerylated and non-phosphoglycerylated sites. A benchmark dataset was employed for this purpose, which contained experimentally identified phosphoglycerylated and non-phosphoglycerylated lysine residues. From the dataset, we extracted the residue adjacency matrix pertaining to each lysine residue in the protein sequences and converted them into feature vectors, which is used to build the phosphoglycerylation predictor. Conclusion: RAM-PGK, which is based on sequential features and support vector machine classifiers, has shown a noteworthy improvement in terms of performance in comparison to some of the recent prediction methods. The performance metrics of the RAM-PGK predictor are: 0.5741 sensitivity, 0.6436 specificity, 0.0531 precision, 0.6414 accuracy, and 0.0824 Mathews correlation coefficient.

In Section 4 of the original paper ‘Complex adjacency matrix and energy of digraphs’, we incorrectly asserted that the iota energy of digraphs increases with respect to the quasi-order rela...

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 $$ RV_{a} $$ shows similarity to EE in ordering of alkanes with 8–10 carbon atoms, whereas index $$ RV_{b} $$ 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.

We study the Wasserstein metric to measure distances between molecules represented by the atom index dependent adjacency 'Coulomb' matrix, used in kernel ridge regression based supervised learning. Resulting machine learning models of quantum properties, a.k.a. quantum machine learning models exhibit improved training efficiency and result in smoother predictions of energies related to molecular distortions. We first illustrate smoothness for the continuous extraction of an atom from some organic molecule. Learning curves, quantifying the decay of the atomization energy's prediction error as a function of training set size, have been obtained for tens of thousands of organic molecules drawn from the QM9 data set. In comparison to conventionally used metrics (L1 and L2 norm), our numerical results indicate systematic improvement in terms of learning curve off-set for random as well as sorted (by norms of row) atom indexing in Coulomb matrices. Our findings suggest that this metric corresponds to a favorable similarity measure which introduces index-invariance in any kernel based model relying on adjacency matrix representations.

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.

An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate that the lower bound is never achieved.

The maximum eigenvalue of the adjacency matrix (AM) has been supposed to contain rich information about the corresponding network. An experimental study focused on revealing the meaning and application of the maximum eigenvalue is missing. To this end, AM was constructed using mutual information (MI) to determine the functional connectivity with electroencephalogram (EEG) data recorded with a mental fatigue model, and then was converted into both binary and weighted brain functional network (BFN) and corresponding random networks (RNs). Both maximum eigenvalue and corresponding network characters in BFNs and RNs were considered to explore the changes during the formation of mental fatigue. The results indicated that large maximum eigenvalue means more edges in the corresponding network, along with a high degree and a short characteristic path length both in weighted and binary BFNs. Interestingly, the maximum eigenvalue of AM was always a little larger than that of the corresponding random matrix (RM), and had an obvious linearity with the sum of the AM elements, indicating that the maximum eigenvalue can be able to distinguish the network structures which have the same mean degree. What is more, the maximum eigenvalue, which increased with the deepening of mental fatigue, can become a good indicator for mental fatigue estimation.

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.

An adjacency matrix A(G) of directed graph G is an m×m matrix consisting of only entries 0 and 1, where m is the number of vertices of G. The entry aij is equal to 1 if there exists a directed edge from vertex vi to vertex vj, otherwise it is equal to 0. Let D(G) be a diagonal matrix of size m×m with each of its main diagonal entry being the degree of the corresponding vertex of directed graph G. Then the matrix L(G) = D(G) − A(G) is called the Laplacian matrix of G. Since a directed graph has two types of degrees namely indegree and outdegree, they result in directed graphs also having the both types of its Laplacian matrices. In this study, the adjacency matrix and the Laplacian matrix of cyclic directed prism graph are investigated. The general form of the coefficients of polynomial characteristic of the Adjacency matrix and Laplacian matrix, respectively is obtained by applying the row reduction method in linear algebra, whereas the general form of the eigenvalues of the polynomial characteristic of the Adjacency matrix and Laplacian matrix, respectively is obtained by factorization and substitution methods.

In this article, we analyze the limiting eigen-value distribution (LED) of random geometric graphs (RGGs). The RGG is constructed by uniformly distributing n nodes on the d-dimensional torus $\Gamma^{d}\equiv [0, 1]^{d}$ and connecting two nodes if their $\ell_{p}$-distance, $ p\in [1,\ \infty]$ is at most r n . In particular, we study the LED of the adjacency matrix of RGGs in the connectivity regime, in which the average vertex degree scales as $\log(n)$ or faster, i.e., $\Omega(\log(n))$. In the connectivity regime and under some conditions on the radius r n , we show that the LED of the adjacency matrix of RGGs converges to the LED of the adjacency matrix of a deterministic geometric graph (DGG) with nodes in a grid as n goes to infinity. Then, for n finite, we use the structure of the DGG to approximate the eigenvalues of the adjacency matrix of the RGG and provide an upper bound for the approximation error. Index Terms--Random geometric graphs, adjacency matrix, limiting eigenvalue distribution, Levy distance.

Semigraph was defined by Sampathkumar as a generalization of a graph. In this paper the adjacency matrix which represents semigraph uniquely and a characterization of such a matrix is obtained. An algorithm to construct the semigraph from a given square matrix, if semigraphical is given. Some properties of adjacency matrix of semigraph are studied. A sufficient condition for eigen values to be real is also obtained.

Let G = (V,E), V = {1,2, …, n}, be a simple graph of order n and size m, without isolated vertices. Denote by d1 ≥ d2 … ≥ dn > 0, di = d(i), a sequence of its vertex degrees. If vertices i and j are adjacent, we write i ~ j. With TI we denote a topological index that can be represented as TI = TI(G) = Σi~j F(di,dj), where F is an appropriately chosen function with the property F(x,y)=F(y,x). Randic vertex-degree-based adjacency matrix RA=(ri j) is defined as rij = Fp(di,dj) √didj, if i ~ j, and 0 otherwise. Denote by f1 ≥ f2 ≥ … ≥ fn the eigenvalues of RA. Upper and lower bounds for fi, i = 1,2, … n are obtained.

This research paper studies about the adjacency matrix of a graph, as it is a fundamental matrix associated with any graph. This study about the properties of adjacency matrix and associated theorems is inevitable since the graph is stored in a computer in terms of its adjacency matrix.