Singular Graph Research Articles

The Singularity of Three Kinds of New Tricyclic Graphs

A graph G is singular if its adjacency matrix is singular. The starting vertices of two paths Pb1 and Pb2 are simultaneously bound to the ending vertex of the path Ps1, and the ending vertices of the paths Pb1 and Pb2 are bound to the starting vertex of path Ps2. Meanwhile, the starting vertex of the path Ps1 is bound to a vertex of the cycle Ca1, and the ending vertex of the path Ps2 is bound to a vertex of the cycle Ca2. Thus, the resulting graph is written as ξ(a1,a2,b1,b2,s1,s2). This is denoted by ζ(a1,a2,b1,b2,s)=ξ(a1,a2,b1,b2,1,s) and ε(a1,a2,b1,b2)=ζ(a1,a2,b1,b2,1), which are referred to as the ξ-graph, ζ-graph and ε-graph for short, respectively. It is known that there are 15 kinds of tricyclic graphs. The purpose of this paper is to study the necessary and sufficient conditions for ξ-graphs, ζ-graphs and ε-graphs to be singular graphs. We analyzed the structure of the elementary spanning subgraphs of the graph G=ξ(a1,a2,b1,b2,s1,s2). By calculating the determinant of the adjacency matrix of the graph G, the necessary and sufficient conditions for the determinant of the graph G to be zero is obtained, and so the necessary and sufficient conditions for graph ξ(a1,a2,b1,b2,s1,s2) to be singular are obtained. As the corollaries, the necessary and sufficient conditions for graphs ζ(a1,a2,b1,b2,s) and ε(a1,a2,b1,b2) to be singular are also obtained.

Research on a Wind Turbine Gearbox Fault Diagnosis Method Using Singular Value Decomposition and Graph Fourier Transform.

Gearboxes operate in challenging environments, which leads to a heightened incidence of failures, and ambient noise further compromises the accuracy of fault diagnosis. To address this issue, we introduce a fault diagnosis method that employs singular value decomposition (SVD) and graph Fourier transform (GFT). Singular values, commonly employed in feature extraction and fault diagnosis, effectively encapsulate various fault states of mechanical equipment. However, prior methods neglect the inter-relationships among singular values, resulting in the loss of subtle fault information concealed within. To precisely and effectively extract subtle fault information from gear vibration signals, this study incorporates graph signal processing (GSP) technology. Following SVD of the original vibration signal, the method constructs a graph signal using singular values as inputs, enabling the capture of topological relationships among these values and the extraction of concealed fault information. Subsequently, the graph signal undergoes a transformation via GFT, facilitating the extraction of fault features from the graph spectral domain. Ultimately, by assessing the Mahalanobis distance between training and testing samples, distinct defect states are discerned and diagnosed. Experimental results on bearing and gear faults demonstrate that the proposed method exhibits enhanced robustness to noise, enabling accurate and effective diagnosis of gearbox faults in environments with substantial noise.

Mosaics for immersed surface-links

The concept of a knot mosaic was introduced by Lomonaco and Kauffman as a means to construct a quantum knot system. The mosaic number of a given knot K is defined as the minimum integer n that allows the representation of K on an n×n mosaic board. Building upon this, the first author and Nelson extended the knot mosaic system to encompass surface-links through the utilization of marked graph diagrams and established both lower and upper bounds for the mosaic number of the surface-links presented in Yoshikawa's table. In this paper, we establish a mosaic system for immersed surface-links by using singular marked graph diagrams. We also provide the definition and discussion on the mosaic number for immersed surface-links.

Hybrid Base Complex: Extract and Visualize Structure of Hex-dominant Meshes.

Hex-dominant mesh generation has received significant attention in recent research due to its superior robustness compared to pure hex-mesh generation techniques. In this work, we introduce the first structure for analyzing hex-dominant meshes. This structure builds on the base complex of pure hex-meshes but incorporates the non-hex elements for a more comprehensive and complete representation. We provide its definition and describe its construction steps. Based on this structure, we present an extraction and categorization of sheets using advanced graph matching techniques to handle the non-hex elements. This enables us to develop an enhanced visual analysis of the structure for any hex-dominant meshes.We apply this structure-based visual analysis to compare hex-dominant meshes generated by different methods to study their advantages and disadvantages. This complements the standard quality metric based on the non-hex element percentage for hex-dominant meshes. Moreover, we propose a strategy to extract a cleaned (optimized) valence-based singularity graph wireframe to analyze the structure for both mesh and sheets. Our results demonstrate that the proposed hybrid base complex provides a coarse representation for mesh element, and the proposed valence singularity graph wireframe provides a better internal visualization of hex-dominant meshes.

On unimodular graphs with a unique perfect matching

A graph is called unimodular if its adjacency matrix has determinant ±1. This article provides a necessary and sufficient condition for a simple connected graph with a unique perfect matching to be unimodular. In particular, we give a complete characterization of bicyclic unimodular graphs with a unique perfect matching. Moreover, the possible values of the determinant of the adjacency matrix of unicyclic, bicyclic, and tricyclic graphs with a unique perfect matching are also provided in this article. For non-bipartite unicyclic graphs with a unique perfect matching, we address the problem of when the inverse of the corresponding adjacency matrix is diagonally similar to a non-negative matrix. A pseudo-unimodular graph is a singular graph whose product of non-zero eigenvalues of the corresponding adjacency matrix is ±1. We supply a necessary and sufficient condition for a singular graph to be pseudo-unimodular.

Prediction of circRNA-MiRNA Association Using Singular Value Decomposition and Graph Neural Networks.

A large number of experimental studies have shown that circRNAs can act as molecular sponges of microRNAs, interacting with miRNAs to regulate gene expression levels, thereby affecting the development of human diseases. Exploring the potential associations between circRNAs and miRNAs can help understand complex disease mechanisms. Considering that biological experiments are time-consuming and labor-intensive, this study proposes a computational model using a graph neural network and singular value decomposition (CMASG) for circRNA-miRNA association prediction. Specifically, graph neural networks are used to learn nonlinear feature representations of nodes, followed by matrix factorization algorithms to learn linear feature representations of nodes, and then combined feature representations learned from different perspectives. Finally, the lightGBM algorithm was used for circRNA-miRNA association prediction. The proposed CMASG model achieved an AUC value of 0.8804. The experimental results demonstrate the superiority and effectiveness of the CMASG model in predicting circRNA-miRNA association tasks.

Singular graphs with dicyclic or semi-dihedral group action

Let Γ be a finite simple graph with adjacency matrix A. Γ is said to be singular if and only if A has an eigenvalue 0. The nullity of Γ, denoted by null, is the multiplicity of the eigenvalue 0 in the spectrum of Γ. In this paper, we completely determine the singularity of graphs Γ for which the dicylic or semi-dihedral group acts transitively and faithfully on V as a group of automorphisms. Moreover, we give formulas to calculate the nullities for such graphs.

The Singularity of Four Kinds of Tricyclic Graphs

A singular graph G, defined when its adjacency matrix is singular, has important applications in mathematics, natural sciences and engineering. The chemical importance of singular graphs lies in the fact that if the molecular graph is singular, the nullity (the number of the zero eigenvalue) is greater than 0, then the corresponding chemical compound is highly reactive or unstable. By this reasoning, chemists have a great interest in this problem. Thus, the problem of characterization singular graphs was proposed and raised extensive studies on this challenging problem thereafter. The graph obtained by conglutinating the starting vertices of three paths Ps1, Ps2, Ps3 into a vertex, and three end vertices into a vertex on the cycle Ca1, Ca2, Ca3, respectively, is denoted as γ(a1,a2,a3,s1,s2,s3). Note that δ(a1,a2,a3,s1,s2)=γ(a1,a2,a3,s1,1,s2), ζ(a1,a2,a3,s)=γ(a1,a2,a3,1,1,s), φ(a1,a2,a3)=γ(a1,a2,a3,1,1,1). In this paper, we give the necessity and sufficiency that the γ−graph, δ−graph, ζ−graph and φ−graph are singular and prove that the probability that a randomly given γ−graph, δ−graph, ζ−graph or φ−graph being singular is equal to 325512,165256,4364, 2132, respectively. From our main results, we can conclude that such a γ−graph(δ−graph, ζ−graph, φ−graph) is singular if at least one cycle is a multiple of 4 in length, and surprisingly, the theoretical probability of these graphs being singular is more than half. This result promotes the understanding of a singular graph and may be promising to propel the solutions to relevant application problems.

Asymptotic Rules of Equilibrium Desingularization

A local bifurcation analysis of a high-dimensional dynamical system dxdt=f(x) is performed using a good deformation of the polynomial mapping P:Cn→Cn. This theory is used to construct geometric aspects of the resolution of multiple zeros of the polynomial vector field P(x). Asymptotic bifurcation rules are derived from Grothendieck’s theory of residuals. Following the Coxeter–Dynkin classification, the singularity graph is constructed. A detailed study of three types of multidimensional mappings with a large symmetry group has been carried out, namely: 1. A linear singularity (behaves similarly to a one-dimensional complex analysis theory); 2. The lattice singularity (generalized the linear and resembling regular crystal growth models); 3. The fan-shaped singularity (can be split radially like nuclear fission and fusion models).

Object-Based Classification Framework of Remote Sensing Images With Graph Convolutional Networks

Object-based image classification (OBIC) on very-high-resolution (VHR) remote sensing (RS) images is utilized in a wide range of applications. Nowadays, many existing OBIC methods only focus on features of each object itself, neglecting the contextual information among adjacent objects and resulting in low classification accuracy. Inspired by a spectral graph theory, we construct a graph structure from objects generated from VHR RS images and propose an OBIC framework based on truncated sparse singular value decomposition and graph convolutional network (GCN) model, aiming to make full use of relativities among objects and produce an accurate classification. Through conducting experiments on two annotated RS image data sets, our framework obtained 97.2% and 66.9% overall accuracy, respectively, in automatic and manual object segmentation circumstances, within a processing time of about 1/100 of convolutional neural network (CNN)-based methods’ training time.

Singlet open-shell conjugated networks generated by singular graphs: Use of Huckel-like models

Singlet open-shell conjugated networks generated by singular graphs: Use of Huckel-like models

Singular graphs with dihedral group action

Let Γ be a simple undirected graph on a finite vertex set and let A be its adjacency matrix. Then Γ is singular if A is singular. The problem of characterizing singular graphs is easy to state but very difficult to resolve in any generality. In this paper we investigate the singularity of graphs for which the dihedral group acts transitively on vertices as a group of automorphisms.

On the fold thickness of graphs

The graph G' obtained from a graph G by identifying two nonadjacent vertices in G having at least one common neighbor is called a 1-fold of G. A sequence G_0, G_1, G_2, ldots , G_k of graphs such that G_0=G and G_i is a 1-fold of G_{i-1} for each i=1, 2, ldots , k is called a uniform k-folding of G if the graphs in the sequence are all singular or all nonsingular. The fold thickness of G is the largest k for which there is a uniform k-folding of G. We show here that the fold thickness of a singular bipartite graph of order n is n-3. Furthermore, the fold thickness of a nonsingular bipartite graph is 0, i.e., every 1-fold of a nonsingular bipartite graph is singular. We also determine the fold thickness of some well-known families of graphs such as cycles, fans and some wheels. Moreover, we investigate the fold thickness of graphs obtained by performing operations on these families of graphs. Specifically, we determine the fold thickness of graphs obtained from the cartesian product of two graphs and the fold thickness of a disconnected graph whose components are all isomorphic.

Right Singular Vector Projection Graphs: Fast High Dimensional Covariance Matrix Estimation under Latent Confounding

SummaryWe consider the problem of estimating a high dimensional p × p covariance matrix Σ, given n observations of confounded data with covariance Σ+ΓΓT, where Γ is an unknown p × q matrix of latent factor loadings. We propose a simple and scalable estimator based on the projection onto the right singular vectors of the observed data matrix, which we call right singular vector projection (RSVP). Our theoretical analysis of this method reveals that, in contrast with approaches based on the removal of principal components, RSVP can cope well with settings where the smallest eigenvalue of ΓTΓ is relatively close to the largest eigenvalue of Σ, as well as when the eigenvalues of ΓTΓ are diverging fast. RSVP does not require knowledge or estimation of the number of latent factors q, but it recovers Σ only up to an unknown positive scale factor. We argue that this suffices in many applications, e.g. if an estimate of the correlation matrix is desired. We also show that, by using subsampling, we can further improve the performance of the method. We demonstrate the favourable performance of RSVP through simulation experiments and an analysis of gene expression data sets collated by the GTEX consortium.

Existence of regular nut graphs for degree at most 11

A nut graph is a singular graph with one-dimensional kernel and corresponding eigenvector with no zero elements. The problem of determining the orders n for which d-regular nut graphs exist was recently posed by Gauci, Pisanski and Sciriha. These orders are known for d <= 4. Here we solve the problem for all remaining cases d <= 11 and determine the complete lists of all d-regular nut graphs of order n for small values of d and n. The existence or non-existence of small regular nut graphs is determined by a computer search. The main tool is a construction that produces, for any d-regular nut graph of order n, another d-regular nut graph of order n + 2d. If we are given a sufficient number of d-regular nut graphs of consecutive orders, called seed graphs, this construction may be applied in such a way that the existence of all d-regular nut graphs of higher orders is established. For even d the orders n are indeed consecutive, while for odd d the orders n are consecutive even numbers. Furthermore, necessary conditions for combinations of order and degree for vertex-transitive nut graphs are derived.

Construction of larger singular and nonsingular graphs using a path

Construction of larger singular and nonsingular graphs using a path

On the isomorphisms between evolution algebras of graphs and random walks

Evolution algebras are non-associative algebras inspired from biological phenomena, with applications to or connections with different mathematical fields. There are two natural ways to define an evolution algebra associated to a given graph. While one takes into account only the adjacencies of the graph, the other includes probabilities related to the symmetric random walk on the same graph. In this work we state new properties related to the relation between these algebras, which is one of the open problems in the interplay between evolution algebras and graphs. On the one hand, we show that for any graph both algebras are strongly isotopic. On the other hand, we provide conditions under which these algebras are or are not isomorphic. For the case of finite non-singular graphs we provide a complete description of the problem, while for the case of finite singular graphs we state a conjecture supported by examples and partial results. The case of graphs with an infinite number of vertices is also discussed. As a sideline of our work, we revisit a result existing in the literature about the identification of the automorphism group of an evolution algebra, and we give an improved version of it.

Remarks on singular Cayley graphs and vanishing elements of simple groups

Let Gamma be a finite graph and let A(Gamma ) be its adjacency matrix. Then Gamma is singular if A(Gamma ) is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we investigate this problem for Cayley graphs mathrm{Cay}(G,H) when G is a finite group and when the connecting set H is a union of conjugacy classes of G. In this situation, the singularity problem reduces to finding an irreducible character chi of G for which sum _{hin H},chi (h)=0. At this stage, we focus on the case when H is a single conjugacy class h^G of G; in this case, the above equality is equivalent to chi (h)=0. Much is known in this situation, with essential information coming from the block theory of representations of finite groups. An element hin G is called vanishing if chi (h)=0 for some irreducible character chi of G. We study vanishing elements mainly in finite simple groups and in alternating groups in particular. We suggest some approaches for constructing singular Cayley graphs.

Detecting neural assemblies in calcium imaging data

BackgroundActivity in populations of neurons often takes the form of assemblies, where specific groups of neurons tend to activate at the same time. However, in calcium imaging data, reliably identifying these assemblies is a challenging problem, and the relative performance of different assembly-detection algorithms is unknown.ResultsTo test the performance of several recently proposed assembly-detection algorithms, we first generated large surrogate datasets of calcium imaging data with predefined assembly structures and characterised the ability of the algorithms to recover known assemblies. The algorithms we tested are based on independent component analysis (ICA), principal component analysis (Promax), similarity analysis (CORE), singular value decomposition (SVD), graph theory (SGC), and frequent item set mining (FIM-X). When applied to the simulated data and tested against parameters such as array size, number of assemblies, assembly size and overlap, and signal strength, the SGC and ICA algorithms and a modified form of the Promax algorithm performed well, while PCA-Promax and FIM-X did less well, for instance, showing a strong dependence on the size of the neural array. Notably, we identified additional analyses that can improve their importance. Next, we applied the same algorithms to a dataset of activity in the zebrafish optic tectum evoked by simple visual stimuli, and found that the SGC algorithm recovered assemblies closest to the averaged responses.ConclusionsOur findings suggest that the neural assemblies recovered from calcium imaging data can vary considerably with the choice of algorithm, but that some algorithms reliably perform better than others. This suggests that previous results using these algorithms may need to be reevaluated in this light.

Singularity-constrained octahedral fields for hexahedral meshing

Despite high practical demand, algorithmic hexahedral meshing with guarantees on robustness and quality remains unsolved. A promising direction follows the idea of integer-grid maps, which pull back the Cartesian hexahedral grid formed by integer isoplanes from a parametric domain to a surface-conforming hexahedral mesh of the input object. Since directly optimizing for a high-quality integer-grid map is mathematically challenging, the construction is usually split into two steps: (1) generation of a surface-aligned octahedral field and (2) generation of an integer-grid map that best aligns to the octahedral field. The main robustness issue stems from the fact that smooth octahedral fields frequently exhibit singularity graphs that are not appropriate for hexahedral meshing and induce heavily degenerate integer-grid maps. The first contribution of this work is an enumeration of all local configurations that exist in hex meshes with bounded edge valence, and a generalization of the Hopf-Poincaré formula to octahedral fields, leading to necessary local and global conditions for the hex-meshability of an octahedral field in terms of its singularity graph. The second contribution is a novel algorithm to generate octahedral fields with prescribed hex-meshable singularity graphs, which requires the solution of a large nonlinear mixed-integer algebraic system. This algorithm is an important step toward robust automatic hexahedral meshing since it enables the generation of a hex-meshable octahedral field.