Matrices Of Graphs Research Articles

The weight distribution of hulls of binary codes from incidence matrices of complete graphs

We characterise the weights of all codewords of the hull of the binary code spanned by rows of an incidence matrix Bn of the complete graph Kn for n ≥ 4 and n even. Among others, we show that every codeword of the hull is the sum of an even number of rows of Bn and that its weight depends on the number of rows required to span it. We also determine the weight enumerator and the MacWilliams identity for the hull.

Development of a manufacturing sustainability index for MSMEs using a structural approach

Micro, small, and medium-sized businesses (MSME) are urged to adopt sustainable manufacturing practices to combat manufacturing-related environmental problems. A sustainability assessment can help industrial organizations to understand the need of lowering their environmental impact. A manufacturing sustainability index (MSI) is proposed in this work by combining environmental sustainability indicators with a structural approach based on graph theory and matrices. Fifteen environmental indicators are identified through literature and weighed using the best-worst method (BWM). Furthermore, graph theory is used to model the inter-dependencies among indicators. A digraph of the indicators is created from which an analogous matrix is developed to generate a manufacturing sustainability assessment function (MSAF) that led to the evaluation of MSI. Two case examples are used to demonstrate the index's utility. To improve the methodology's applicability, the study produced the Ideal MSI and Manufacturing Sustainability Index Ratio (MSIR). Sensitivity analysis is performed to assess the methodology's robustness and to pinpoint the significant areas of concern. The proposed methodology will assist industry professionals, practitioners, and managers in developing strategies and implementing technologies and practices that will lead to environmentally sustainable manufacturing.

Codes from the incidence matrices of a zero-divisor graphs

In this paper, we examine the linear codes with respect to the Hamming metric from incidence matrices of the zero-divisor graphs with vertex set consisting of all non-zero zero- divisors of the ring ℤ n and two distinct vertices being adjacent if their product is zero over ℤ n . The main parameters of the codes are obtained.

Workflow performance prediction based on graph structure aware deep attention neural network

• In workflow performance prediction, DAG structure matters; • DAG-Transformer effectively embeds the DAG information and outperforms mainstream ML, DL , and GCN methods; • A new dataset for cloud workflow performance prediction is accompanied as well as the source code. With the rapid growth of cloud computing, efficient operational optimization and resource scheduling of complex cloud business processes rely on real-time and accurate performance prediction. Previous research on cloud computing performance prediction focused on qualitative (heuristic rules), model-driven, or coarse-grained time-series prediction, which ignore the study of historical performance, resource allocation status and service sequence relationships of workflow services. There are even fewer studies on prediction for workflow graph data due to the lack of available public datasets. In this study, from Alibaba Cloud's Cluster-trace-v2018, we extract nearly one billion offline task instance records into a new dataset, which contains approximately one million workflows and their corresponding directed acyclic graph (DAG) matrices. We propose a novel workflow performance prediction model (DAG-Transformer) to address the aforementioned challenges. In DAG-Transformer, we design a customized position encoding matrix and an attention mask for workflows, which can make full use of workflow sequential and graph relations to improve the embedding representation and perception ability of the deep neural network. The experiments validate the necessity of integrating graph-structure information in workflow prediction. Compared with mainstream deep learning (DL) methods and several classic machine learning (ML) algorithms, the accuracy of DAG-Transformer is the highest. DAG-Transformer can achieve 85-92% CPU prediction accuracy and 94-98% memory prediction accuracy, while maintaining high efficiency and low overheads. This study establishes a new paradigm and baseline for workflow performance prediction and provides a new way for facilitating workflow scheduling.

Multi-view unsupervised feature selection with tensor low-rank minimization

To describe objects more comprehensively and accurately, multi-view learning has attracted considerable attention. Recently, graph embedding based multi-view feature selection methods have been proposed and shown efficient in many real applications. The existing methods generally construct one common graph matrix to exploit the local structure of multi-view data via the linear weight fusion or learning one common graph matrix across all views. However, since all views share the identical graph structure, this emphasizes the consistency too much, resulting in restricting the diversity among different views. In this paper, a tensor low-rank constrained graph embedding method is proposed for multi-view unsupervised feature selection. To embody the view-specific information of each view, our model constructs the graph structure for each corresponding view, respectively. To capture the consistency across views, a tensor low-rank regularization constraint is imposed on the tensor data formed by these graph matrices. An efficient optimization algorithm with theoretical convergence guarantee is designed to solve the proposed method. Extensive experimental results validate that the proposed method outperforms some state-of-the-art methods. The code of our model can be found athttps://www.researchgate.net/publication/353902948_demoTLR.

Initial state estimation from limited observations of the heat equation in metric graphs

This paper deals with initial state estimation problems of the heat equation in equilateral metric graphs being admitted to have cycles. Particularly, we are concerned with suitable placements of observation points in order to uniquely determine the initial state from observation data. We give a necessary and sufficient condition for suitable placements of observation points, and such suitable placements are determined from transition matrices of metric graphs. From numerical simulations, we confirm effectiveness of a necessary and sufficient condition.

Dynamic Graph Convolution Network for Traffic Forecasting Based on Latent Network of Laplace Matrix Estimation

Traffic forecasting is a challenging problem in the transportation research field as the complexity and non-stationary changing of the traffic data, thus the key to the issue is how to explore proper spatial and temporal characteristics. Based on this thought, many creative methods have been proposed, in which Graph Convolution Network (GCN) based methods have shown promising performance. However, these methods depend on the graph construction, which mainly uses the prior knowledge of the road network. Recently, some works realized the fact of the road network graph changing and tried to construct dynamic graphs for GCN, but they do not fully exploit the spatial and temporal properties of the traffic data in the graph construction. In this paper, we propose a novel dynamic graph convolution network for traffic forecasting, in which a latent network is introduced to extract spatial-temporal features for constructing the dynamic road network graph matrices adaptively. The proposed method is evaluated on several traffic datasets and the experimental results show that it outperforms the state of the art traffic forecasting methods. The website of the code is <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/guokan987/DGCN.git</uri> .

Real symmetric matrices and their negative eigenvalues

For two integers k≧0 and q≧1, consider symmetric matrices M with k negative eigenvalues counted with multiplicities and q pairwise distinct values of entries such that the rows of M are mutually distinct and the largest diagonal entry of M is less than or equal to the smallest off-diagonal entry of M. It is shown that the number of such matrices is finite when k and q are fixed. This generalizes some known results on the adjacency matrices of graphs. It is conjectured that any twin-free graph on n vertices with no isolated vertices has at least −1+log2⁡(n+2) negative adjacency eigenvalues.

Real spectra of large real asymmetric random matrices.

When a randomness is introduced at the level of real matrix elements, depending on its particular realization, a pair of eigenvalues can appear as real or form a complex conjugate pair. We show that in the limit of large matrix size the density of such real eigenvalues is proportional to the square root of the asymptotic density of complex eigenvalues continuated to the real line. This relation allows one to calculate the real densities up to a normalization constant, which is then applied to various examples, including heavy-tailed ensembles and adjacency matrices of sparse random regular graphs.

Spectra of Variants of Distance Matrices of Graphs and Digraphs: A Survey

Distance matrices of graphs were introduced by Graham and Pollack in 1971 to study a problem in communications. Since then, there has been extensive research on the distance matrices of graphs—a 2014 survey by Aouchiche and Hansen on spectra of distance matrices of graphs lists more than 150 references. In the last 10 years, variants such as the distance Laplacian, the distance signless Laplacian, and the normalized distance Laplacian matrix of a graph have been studied. After a brief description of the early history of the distance matrix and its motivating problem, this survey focuses on comparing and contrasting techniques and results for the four types of distance matrices. Digraphs are treated separately after the discussion of graphs, including discussion of similarities and differences between graphs and digraphs. New results are presented that complement existing results, including results for some the matrices on unimodality of characteristic polynomials for graphs, preservation of parameters by cospectrality for graphs, and bounds on spectral radii for digraphs.

Eccentricity energy change of complete multipartite graphs due to edge deletion

Abstract The eccentricity matrix ɛ(G) of a graph G is obtained from the distance matrix of G by retaining the largest distances in each row and each column, and leaving zeros in the remaining ones. The eccentricity energy of G is sum of the absolute values of the eigenvalues of ɛ(G). Although the eccentricity matrices of graphs are closely related to the distance matrices of graphs, a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The change in eccentricity energy of a graph due to an edge deletion is one such property. In this article, we give examples of graphs for which the eccentricity energy increase (resp., decrease) but the distance energy decrease (resp., increase) due to an edge deletion. Also, we prove that the eccentricity energy of the complete k-partite graph Kn 1, ... , nk with k ≥ 2 and ni ≥ 2, increases due to an edge deletion.

On the spectral radius and the energy of eccentricity matrices of graphs

The eccentricity matrix of a connected graph G is obtained from the distance matrix of G by retaining the largest distance in each row and each column and setting the remaining entries as 0. The eccentricity matrices of graphs are closely related to the distance matrices of graphs, nevertheless a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The eccentricity matrices of trees need not be invertible though the distance matrices of trees are invertible. In this paper, we establish a necessary and sufficient condition for the eccentricity matrix of a tree to be invertible. The largest eigenvalue of is called the ε-spectral radius, and the eccentricity energy (or the ε-energy) of G is the sum of the absolute values of the eigenvalues of . We obtain some bounds for the ε-spectral radius and characterize the extreme graphs. Two graphs are said to be ε-equienergetic if they have the same ε-energy. For any , we construct a pair of ε-equienergetic graphs on n vertices, which are not ε-cospectral.

A Multi-Factorial Assessment of Functional Human Autistic Spectrum Brain Network Analysis.

The variability of the results obtained by the statistical analysis of functional human brain networks depend on multiple factors such as: the source of the fMRI data, the brain parcellations, the graph theory measures, and the threshold values applied to the functional connectivity matrices to obtain adjacency matrices of sparse graphs. Therefore, the brain network used for down-stream analysis is heavily dependent on the methods that are applied to the fMRI data to obtain and analyze such networks. In this paper we present the preliminary results of a multi-factorial assessment of the statistical analysis of functional human brain networks. The assessment was performed in the functional human brain networks obtained from the resting state fMRI data of ten imaging sites provided by the Autism Brain Imaging Data Exchange (ABIDE) preprocessed functional magnetic resonance database, with six different functional brain parcellations, six different graph theory measures, and three different threshold values applied to the corresponding connectivity matrices to obtain sparse graphs. The statistical analysis to detect differences between the networks representing autism and control subjects were performed with four different statistical methods, using the p-values to determine the levels of significance of the analysis. Our main results show a strong dependence of functional human brain networks statistical analysis on the brain parcellations, and on the graph theory measures. Our results further show that the results of these analysis are less dependent on the statistical tests methods and on the threshold values of the sparse graphs for all practical purposes. An additional result is that the levels of significance of the statistical tests obtained for data provided by individual sites were much higher than the global levels of significance obtained by averaging the results of all the sites, implying that the best results on the analysis of functional human brain networks are obtained when the source of the fMRI data is the same for all the data. Since reproducibility and reliability of functional brain network statistical analysis is strongly dependent on the graphs obtained from fMRI data; our expectation is that the novel results presented in this paper would further help researchers in this field to develop methods that are reliable and reproducible.

On the Ger\v{s}gorin disks of distance matrices of graphs

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)=Tr(G)-D(G)$, and $D^{Q}(G)=Tr(G)+D(G)$ be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix, and the distance signless Laplacian matrix of $G$, respectively. Atik and Panigrahi [2] suggested the study of the problem: Whether all eigenvalues, except the spectral radius, of $ D(G) $ and $ D^{Q}(G) $ lie in the smallest Ger\v{s}gorin disk? In this paper, we provide a negative answer by constructing an infinite family of counterexamples.

Invertibility of adjacency matrices for random d-regular graphs

Let d≥3 be a fixed integer, and let A be the adjacency matrix of a random d-regular directed or undirected graph on n vertices. We show that there exists a constant d>0 such that P(A is singular in R)≤n−d, for n sufficiently large. This answers an open problem by Frieze and Vu. The key idea is to study the singularity probability over a finite field Fp. The proof combines a local central limit theorem and a large deviation estimate.

Linear codes from incidence matrices of unit graphs

In this paper, we examine the binary linear codes with respect to Hamming metric from incidence matrix of a unit graph G(ℤ n ) with vertex set is ℤ n and two distinct vertices x and y being adjacent if and only if x + y is unit. The main parameters of the codes are given.

A Characteristic Polynomial for the Transition Probability Matrix of Correlated Random Walks on a Graph

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As an application, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph.

Spatial‐temporal slowfast graph convolutional network for skeleton‐based action recognition

In skeleton-based action recognition, the graph convolutional network (GCN) has achieved great success. Modelling skeleton data in a suitable spatial-temporal way and designing the adjacency matrix are crucial aspects for GCN-based methods to capture joint relationships. In this study, we propose the spatial-temporal slowfast graph convolutional network (STSF-GCN) and design the adjacency matrices for the skeleton data graphs in STSF-GCN. STSF-GCN contains two pathways: (1) the fast pathway is in a high frame rate, and joints of adjacent frames are unified to build ‘small’ spatial-temporal graphs. A new spatial-temporal adjacency matrix is proposed for these ‘small’ spatial-temporal graphs. Ablation studies verify the effectiveness of the proposed adjacency matrix. (2) The slow pathway is in a low frame rate, and joints from all frames are unified to build one ‘big’ spatial-temporal graph. The adjacency matrix for the ‘big’ spatial-temporal graph is obtained by computing self-attention coefficients of each joint. Finally, outputs from two pathways are fused to predict the action category. STSF-GCN can efficiently capture both long-range and short-range spatial-temporal joint relationships. On three datasets for skeleton-based action recognition, STSF-GCN can achieve state-of-the-art performance with much less computational cost.

New production matrices for geometric graphs

We use production matrices to count several classes of geometric graphs. We present novel production matrices for non-crossing partitions, connected geometric graphs, and k-angulations, which provide another, simple and elegant, way of counting the number of such objects. Counting geometric graphs is then equivalent to calculating the powers of a production matrix. Applying the technique of Riordan Arrays to these production matrices, we establish new formulas for the numbers of geometric graphs as well as combinatorial identities derived from the production matrices. Further, we obtain the characteristic polynomial and the eigenvectors of such production matrices.

Gap sets for the spectra of cubic graphs

We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals ( 2 2 , 3 ) (2 \sqrt {2},3) and [ − 3 , − 2 ) [-3,-2) achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in [ − 3 , 3 ] [-3,3] which are maximally gapped and construct examples which achieve these bounds. These graphs yield new instances of maximally gapped intervals. We also show that every point in [ − 3 , 3 ) [-3,3) can be gapped by planar cubic graphs. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.