Matrices Of Graphs Research Articles

Bottleneck matrices of maximal outerplanar graphs with isomorphic underlying trees

In this paper, we consider the entries of the bottleneck matrices of maximal outerplanar graphs with isomorphic underlying trees. We show how the entries of the bottleneck matrix are perturbed when we modify a maximal outerplanar graph into a nonisomorphic maximal outerplanar graph whose underlying tree is isomorphic to the underlying tree of the original maximal outerplanar graph. We illustrate our results with examples.

TAKAGI–SUGENO–KANG FUZZY SYSTEM MODELING BASED ON LOW-RANK SPARSE SUBSPACE LEARNING FOR MOTOR IMAGERY ELECTROENCEPHALOGRAM SIGNAL CLASSIFICATION

The classification of electroencephalogram (EEG) signals derived from motor imagery (MI) has always been a hot topic in the field of brain–computer interfaces. Due to its ability to handle the nonstationary and uncertain information contained in EEG signals, the Takagi–Sugeno–Kang fuzzy system (TSK-FS) has become an advantageous classification algorithm. To train a fuzzy system with strong discrimination capabilities from EEG data interspersed with redundant information, this paper proposes a TSK-FS modeling method based on low-rank sparse subspace learning (TSK-LSSL). This method focuses on consequent parameter learning, which transforms the traditional consequent parameter learning strategy into low-rank subspace and sparse subspace learning processes. Low-rank subspace learning is used to mine the global structural information of data and effectively reduce the number of fuzzy rules. During sparse subspace learning, [Formula: see text]-norm regularization is used to constrain the consequent parameters and causes the number of redundant consequent parameters to be zero, thereby simplifying the fuzzy rules. In addition, a local boundary term based on graph matrices is embedded into the objective function to mine the local structural information of the given data. TSK-LSSL simplifies the number of rules and the consequent part of the fuzzy rules. It exhibits good classification performance on two BCI Competition databases.

Entries of the bottleneck matrices of maximal outerplanar graphs

In this paper, we consider the entries of the bottleneck matrices of maximal outerplanar graphs. We recall patterns from [Kirkland SJ, Neumann M, Shader BL. Characteristic vertices of weighted trees via Perron values. Linear Multilinear Algebra. 1996;40:311–325.] involving the entries of bottleneck matrices for trees. We first establish similar patterns involving the diagonal entries of bottleneck matrices for maximal outerplanar graphs. We then investigate patterns involving the off-diagonal entries of bottleneck matrices for maximal outerplanar graphs and establish similar results to those found in [Kirkland SJ, Neumann M, Shader BL. Characteristic vertices of weighted trees via Perron values. Linear Multilinear Algebra. 1996;40:311–325.] for trees. We close with examples illustrating our results.

On Spectral Portraits of Incidence Matrices of Nearest Neighbor Graphs

On Spectral Portraits of Incidence Matrices of Nearest Neighbor Graphs

Exploration of regularities in bipartite graphs using GEOGEBRA software

A classroom proposal is presented to integrate contents of Graph Theory and Linear Algebra in complete bipartite graphs, linking adjacency and Laplacian matrices, the eigenvalues of graphs will be determined, applicable to connectivity concepts. Students will be given exploration activities working with GeoGebra software, starting from several particular cases, with table works and questionnaires to be completed, in order to determine patterns on the eigenvalues of adjacency and Laplacian matrices of complete bipartite graphs. The work with patterns will lead to the generalization process, to abstract properties from observation and experimentation on examples. This learning experience builds bridges between the concrete and the symbolic, and the student is initiated in research

Limit points of (signless) Laplacian spectral radii of linear trees

We study limit points of the spectral radii of Laplacian matrices of graphs. We adapted the method used by J. B. Shearer in 1989, devised to prove the density of adjacency limit points of caterpillars, to Laplacian limit points. We show that this fails, in the sense that there is an interval for which the method produces no limit points. Then we generalize the method to Laplacian limit points of linear trees and prove that it generates a larger set of limit points. The results of this manuscript may provide important tools for proving the density of Laplacian limit points in [4.38+,∞).

Unified approach for spectral properties of weighted adjacency matrices for graphs with degree-based edge-weights

Let G be a graph and di be the degree of a vertex vi in G. For a symmetric real function f(x,y), one can get an edge-weighted graph in such a way that for each edge vivj of G, the weight of vivj is assigned by f(di,dj). Hence, we have a weighted adjacency matrix Af(G) of G, in which the ij-entry is equal to f(di,dj) if vivj∈E(G) and 0 otherwise. In this paper, we use a unified approach to deal with the spectral properties of Af(G) for f(x,y) to be the functions of graphical or topological function-indices. Firstly, we obtain uniform interlacing inequalities for the weighted adjacency eigenvalues. For the edge-weight functions defined by almost a half of popularly used topological indices, it can be shown that our inequalities cannot be improved. Secondly, we establish a uniform equivalent condition for a connected graph G to have m distinct weighted adjacency eigenvalues. As an application, a combinatorial characterization for a graph to have two and three distinct weighted adjacency eigenvalues is presented, respectively. Moreover, bipartite graphs and unicyclic graphs with three distinct weighted adjacency eigenvalues are characterized. This paper attempts to unify the spectral study for weighted adjacency matrices of graphs with degree-based edge-weights.

New qubit codes from multidimensional circulant graphs

Two new qubit stabilizer codes with parameters 〚77,0,19〛2 and 〚90,0,22〛2 are constructed for the first time by employing additive symplectic self-dual F4 codes from multidimensional circulant (MDC) graphs. We completely classify MDC graph codes for lengths 4≤n≤40 and show that many optimal 〚ℓ,0,d〛 qubit codes can be obtained from the MDC construction. Moreover, we prove that adjacency matrices of MDC graphs have nested block circulant structure and determine isomorphism properties of MDC graphs.

Optimizing energy storage plant discrete system dynamics analysis with graph convolutional networks

Addressing the challenges of suboptimal model performance and excessive parameters and operations in the optimization of energy storage power plants utilizing Graph Convolutional Network (GCN), this paper introduces a novel approach – the packet-switched graph convolutional network. Initially, a GCN extreme learning machine is established. Drawing inspiration from this solid foundation, we have innovatively crafted a group exchange graph convolution module. This module leverages group graph convolution techniques to amalgamate unique node feature information, tailored to diverse topology graph matrices based on various groupings. This innovative approach ensures that information flows freely and effectively among distinct groupings. Furthermore, we have designed a cutting-edge timing depth separation convolution module, comprising two innovative components. The first component introduces timing depth separation convolution, revolutionizing the original timing convolution module. The second component, the packet-switching graph convolutional network, revolutionizes the time sequence depth separation convolution process. It achieves this by employing 1 × 1 convolutional layers between different feature fusion packets, enabling seamless information exchange between distinct packets. Experimental results demonstrate the efficacy of the proposed model, with root mean square error (RMSE) metrics and root mean square error (MAE) metrics for single-step prediction reaching 46.08 and 26.22 at 60 min, respectively. In multi-step testing, the proposed model exhibits a 14.71 % reduction in RMSE error at the 15-min scale and a 9.29 % reduction at the 60-min scale compared to the benchmark model. This performance improvement enhances the operational efficiency and reliability of the energy storage plant, particularly under dynamic changes in the time series.

Characteristic Polynomial of Power Graph for Dihedral Groups Using Degree-Based Matrices

A fundamental feature of spectral graph theory is the correspondence between matrix and graph. As a result of this relation, the characteristic polynomial of the graph can be formulated. This research focuses on the power graph of dihedral groups using degree-based matrices. Throughout this paper, we formulate the characteristic polynomial of the power graph of dihedral groups based on seven types of graph matrices which include the maximum degree, the minimum degree, the greatest common divisor degree, the first Zagreb, the second Zagreb, the misbalance degree, and the Nirmala matrices.

The Index Weighted Hermitian Adjacency Matrices for Mixed Graphs

Based on the Hermitian adjacency matrices of second kind introduced by Mohar [1] and weighted adjacency matrices introduced in [2], we define a kind of index weighted Hermitian adjacency matrices of mixed graphs. In this paper we characterize the structure of mixed graphs which are cospectral to their underlying graphs, then we determine a upper bound on the spectral radius of mixed graphs with maximum degree Δ , and characterize the corresponding extremal graphs.

Rational orthogonal matrices and isomorphism of graphs

Let On(Q) and Sn(Z) be the set of all n by n orthogonal matrices with rational entries and the set of all n by n symmetric matrices with integer entries, respectively. We consider the following question: Given A∈Sn(Z). When does QTAQ∈Sn(Z) with Q∈On(Q) imply that Q is a signed permutation matrix? Wang and Yu [15] show that the answer to the above question is yes whenever the discriminant Δ(A) of the characteristic polynomial of A is odd and square-free. However, the above result cannot be applied for the adjacency matrix of a simple graph, since Δ(A) can never be odd in this situation. We give a new condition under which the answer is still yes for the adjacency matrices of graphs. The result was achieved by employing some tools from our recent work on the generalized spectral characterizations of graphs, which is somewhat unexpected. Some applications of the above result are also provided.

About the Data Analysis of Optical Emission Spectra of Reactive Ion Etching Processes—The Method of Spectral Redundancy Reduction

In this study, we present the Method of Spectral Redundancy Reduction (MSRR) for analyzing OES (optical emission spectroscopy) data of dry etching processes based on the principles of spectral clustering. To achieve this, the OES data are transformed into abstract graph matrices whose associated eigenvectors directly indicate anomalies in the data set. We developed an approach that allows for the reduction in temporally resolved optical emission spectra from plasma structuring processes in such a way that individual emission lines can be algorithmically detected, which exhibit a temporal behavior different from the collective behavior of the temporally resolved overall spectrum. The proportion of emission lines that behave consistently throughout the entire process duration is not considered. Our work may find applications in which OES is used as a process-monitoring technique, especially for low-pressure plasma processing. The major benefit of the developed method is that the scale of the original data is kept, making physical interpretations possible despite data reductions.

Absent Multiview Semisupervised Classification.

With the advent of vast data collection ways, data are often with multiple modalities or coming from multiple sources. Traditional multiview learning often assumes that each example of data appears in all views. However, this assumption is too strict in some real applications such as multisensor surveillance system, where every view suffers from some data absent. In this article, we focus on how to classify such incomplete multiview data in semisupervised scenario and a method called absent multiview semisupervised classification (AMSC) has been proposed. Specifically, partial graph matrices are constructed independently by anchor strategy to measure the relationships among between each pair of present samples on each view. And to obtain unambiguous classification results for all unlabeled data points, AMSC learns view-specific label matrices and a common label matrix simultaneously. AMSC measures the similarity between pair of view-specific label vectors on each view by partial graph matrices, and consider the similarity between view-specific label vectors and class indicator vectors based on the common label matrix. To characterize the contributions of different views, the p th root integration strategy is adopted to incorporate the losses of different views. By further analyzing the relation between the p th root integration strategy and exponential decay integration strategy, we develop an efficient algorithm with proved convergence to solve the proposed nonconvex problem. To validate the effectiveness of AMSC, comparisons are made with some benchmark methods on real-world datasets and in the document classification scenario as well. The experimental results demonstrate the advantages of our proposed approach.

Structured discriminative Gaussian graph learning for multimode process monitoring

AbstractAiming at the actual industrial process background that different modes share the same system configurations and control structure, this article proposes a novel structured discriminant Gaussian graph learning for multimode process monitoring. The proposed method considers not only the sparsity of graph model but also the measurement of data variation based on a mismatched graph and the common node support between different graphical structures. The objective function involves two sets of regularization terms: the trace terms for mismatched measurements and the ‐norm imposed on the union of decomposed graph matrices. Due to the introduced mismatched trace terms, the cost of matching the data points and graph models that have inconsistent class labels can be expanded, which brings more discrimination for the graph‐based mode identification. While the common structure extracted by the ‐norm forces the estimated graph models to have structural similarities, thus alleviating the negative influence caused by graph discrimination. Once a relatively accurate and discriminative reference graph model is obtained, the downstream test graph learning and analysis can be conducted online by employing the moving window techniques. By comparing the matched and mismatched graph‐based measurements, the process mode can be identified correctly and stably. To grasp the abnormal process changes, the ‐norm for the row sparsity is again applied to the graph difference matrices, the sensitive monitoring statistics and the fault isolation results can be obtained effectively. All the optimization problems in this paper can be solved using the alternating direction multiplier (ADMM) algorithm. The effectiveness of our proposed approach is illustrated by the application to a real blast furnace iron‐making production process.

Distance matrices for conjugate skew gain graphs

A conjugate skew gain graph is a skew gain graph with the labels (also called, the conjugate skew gains) from the field of complex numbers on the oriented edges such that they get conjugated when we reverse the orientation. In this paper, we introduce distance matrices for conjugate skew gain graphs and characterize balanced conjugate skew gain graphs using these matrices. We provide explicit formulae for the distance spectra of certain conjugate skew gain graphs.

Universal Distance Spectra of Join of Graphs

Consider G a simple connected graph. In this paper, we introduce the Universal distance matrix UD (G). For α, β, γ, δ ∈ R and β ̸= 0, the universal distance matrix UD (G) is defined asUD (G) = αTr (G) + βD(G) + γJ + δI,where Tr (G) is the diagonal matrix whose elements are the vertex transmissions, and D(G) is the distance matrix of G. Here J is the all-ones matrix, and I is the identity matrix. In this paper, we obtain the universal distance spectra of regular graph, join of two regular graphs, joined union of three regular graphs, generalized joined union of n disjoint graphs with one arbitrary graph H. As a consequence, we obtain the eigenvalues of distance matrix, distance Laplacian matrix, distance signless Laplacian matrix, generalized distance matrix, distance Seidal matrix and distance matrices of complementary graphs.

Euclidean and circum-Euclidean distance matrices: characterizations and interlacing property

Motivated by the inverse formula of the distance matrix of a tree and the Moore–Penrose inverse of a circum-Euclidean distance matrix (CEDM), in this paper, we study a general real square matrix M whose Moore–Penrose inverse can be expressed as the sum of a Laplacian-like matrix L and a rank one matrix. In particular, for a symmetric hollow matrix M, under an assumption, we show that M is a Euclidean distance matrix if and only if L is positive semidefinite. Based on this, we obtain a new characterization for CEDMs involving their Moore–Penrose inverses. As an application, we show that the distance matrices of block graphs and odd-cycle-clique graphs are CEDMs. Finally, we establish an interlacing property between the eigenvalues of a Euclidean distance matrix M (including the singular case) and its associated Laplacian-like matrix L, which generalizes the interlacing property proved for the distance matrices of trees.

Equivalence of lattice operators and graph matrices

Abstract We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix called an “antisymmetrized adjacency matrix”, specifically tailored for cycle digraphs (T1 lattice) and simple directed paths (B1 lattice). The nontrivial relationship between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the antisymmetrized adjacency matrix, along with its extensions to higher dimensions, is equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D-dimensional graph results from a Cartesian product of cycle digraphs (T1 lattice) and simple directed paths (B1 lattice).

On the Energy and Spread of the Adjacency, Laplacian and Signless Laplacian Matrices of Graphs

In this paper, we explore the connection between the energy and spread of the adjacency, Laplacian, and signless Laplacian matrices for graphs. We then introduce new limitations for the energy and spread of these matrices, based on previous research and our findings.