Spectral Graphs Research Articles

Spectral extremal graphs for edge blow-up of star forests

The edge blow-up of a graph G, denoted by Gp+1, is obtained by replacing each edge of G with a clique of order p+1, where the new vertices of the cliques are all distinct. Yuan (2022) [20] determined the range of the Turán numbers for edge blow-up of all bipartite graphs and the exact Turán numbers for edge blow-up of all non-bipartite graphs. In this paper, we prove that the graphs with the maximum spectral radius in an n-vertex graph without any copy of edge blow-up of star forests are the extremal graphs for edge blow-up of star forests when n is sufficiently large.

Electric Motor by using Machine Learning.

This research paper addresses the crucial issue of early bearing failure within electric motors, a common problem impacting industrial operations. It begins with a thorough problem identification and historical study of asset failures to pinpoint strategic sensor placement on the motor casing, specifically targeting the bearing area. Time series data acquisition captures the dynamic motor behavior, with LabVIEW software facilitating visualization through frequency plots and spectral graphs. Subsequent preprocessing involves modifying data structure and computing vibration metrics like RRMS and IRRMS to quantify vibration levels. The processing stage consists of condition monitoring using structured supervised datasets transformed into the frequency domain in MATLAB. Interpretation of results enables bearing health assessment and identification of dominant frequency components. Additionally, the remaining useful life of the bearing is estimated using the Support Vector Classifier algorithm, supplemented by RRMS and IRRMS to enhance prediction accuracy. A machine learning model, developed in Python and trained on bearing datasets, predicts the remaining useful life based on sensor-collected data. This comprehensive methodology aims to provide insights into bearing health, facilitating proactive maintenance and optimizing industrial operations

Spectral Extremal Results on Trees

Let ${\rm spex}(n,F)$ be the maximum spectral radius over all $F$-free graphs of order $n$, and ${\rm SPEX}(n,F)$ be the family of $F$-free graphs of order $n$ with spectral radius equal to ${\rm spex}(n,F)$. Given integers $n,k,p$ with $n>k>0$ and $0\leq p\leq \lfloor(n-k)/2\rfloor$, let $S_{n,k}^{p}$ be the graph obtained from $K_k\nabla(n-k)K_1$ by embedding $p$ independent edges within its independent set, where '$\nabla$' means the join product. For $n\geq\ell\geq 4$, let $G_{n,\ell}=S_{n,(\ell-2)/2}^{0}$ if $\ell$ is even, and $G_{n,\ell}=S_{n,(\ell-3)/2}^{1}$ if $\ell$ is odd. Cioabă, Desai and Tait [SIAM J. Discrete Math. 37 (3) (2023) 2228-2239] showed that for $\ell\geq 6$ and sufficiently large $n$, if $\rho(G)\geq \rho(G_{n,\ell})$, then $G$ contains all trees of order $\ell$ unless $G=G_{n,\ell}$. They further posed a problem to study ${\rm spex}(n,F)$ for various specific trees $F$. Fix a tree $F$ of order $\ell\geq 6$, let $A$ and $B$ be two partite sets of $F$ with $|A|\leq |B|$, and set $q=|A|-1$. We first show that any graph in ${\rm SPEX}(n,F)$ contains a spanning subgraph $K_{q,n-q}$ for $q\geq 1$ and sufficiently large $n$. Consequently, $\rho(K_{q,n-q})\leq {\rm spex}(n,F)\leq \rho(G_{n,\ell})$, we further respectively characterize all trees $F$ with these two equalities holding. Secondly, we characterize the spectral extremal graphs for some specific trees and provide asymptotic spectral extremal values of the remaining trees. In particular, we characterize the spectral extremal graphs for all spiders, surprisingly, the extremal graphs are not always the spanning subgraph of $G_{n,\ell}$.

Fire zone forefront residual tar combustion characteristics and functional group transformation mechanism: Appling the based on thermogravimetric and in-situ high-temperature FTIR

When preventing spontaneous combustion disasters in extinguished coal fire zones, the influence of the tar re-ignition phenomenon is often overlooked. Therefore, in this study, using thermogravimetric analysis and high-temperature (630 °C) in-situ infrared technology, the combustion reactivity and real-time evolution mechanism of key functional groups for tar were deeply discussed under oxygen-lean conditions. Additionally, a novel two-step baseline drift correction method for spectral graphs at high temperatures is proposed. The results showed that tar combustion could be divided into two stages: pyrolysis-volatilization mass loss and crosslinked structure oxidation combustion. Reduced oxygen concentration limits the violent and stable rapid combustion of tar-crosslinked structures and reduces the re-ignition tendency of tar. Among, with a decrease in oxygen concentration, H (burnout performance), Cb (the reaction ability in the early stage of ignition) and S (overall combustion performance) decreased, while Ti (mass loss endpoint temperature), Th(burnout temperature), and HF (combustion stability) increased. This provides information for the selection of a reasonable process to prevent tar re-ignition. The contents of methylidene and aldehyde carbonyl in tar increased by approximately 2 and 3.7 times, respectively, compared to those of raw coal. In the first thermal decomposition stage of tar, the aromatic hydrocarbons were relatively stable under the action of the structural framework, whereas aliphatic hydrocarbons continued to decrease. Before burnout, a decrease in oxygen concentration increased the crosslinking ability of tar and delayed the tar burnout process. The contents of aromatic hydrocarbons and aldehyde groups are highly sensitive to the evolution of the oxygen concentration. Specifically, the Kubelka–Munk value of the aromatic hydrocarbons increased from 0.015 to 0.10, while that of aldehyde carbonyl groups increased from 0 to 0.23. Additionally, in the early stage of tar combustion, the volatile mass loss was basically not affected by the time-scale effect, and mass loss endpoint residual mass was approximately 48.11 %. In the later stage of tar combustion, owing to the cumulative of time-scale effect, the tar re-ignition tendency increased, H, Cb, and S increased, and HF decreased from 1.388 to 1.098. This research is helpful guiding disaster prevention and control of tar re-ignition in a fire area, which leads to reburning in the surrounding area.

A research on an improved fuzzy approximate entropy algorithm for EMG-based shoulder and neck muscle fatigue detection

A robust muscle fatigue algorithm plays a pivotal role in depicting the degree of muscle fatigue in both time-series EMG signal graphs and spectral graphs, aligning with human perception. While the fuzzy approximate entropy (fApEn) algorithm has been enhanced from the foundation of approximate entropy (ApEn) through the incorporation of fuzzy affiliation, concerns persist regarding the threshold value and the algorithm’s application range. This study extracts EMG signals across varied time durations and head-down angles, employing enhanced signal preprocessing techniques and optimizing the fApEn algorithm. Furthermore, real-time fatigue perceptions of subjects were recorded using the rating of perceived exertion. Experimental outcomes reveal that the EMG signal, post-wavelet analysis preprocessing, demonstrates promising noise reduction capabilities. Notably, the fApEn algorithm exhibits considerable enhancements through the identification of an optimal threshold using the gradient descent algorithm and a machine learning strategy.

A Spectral Extremal Problem on Non-Bipartite Triangle-Free Graphs

A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda(G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. A well-known spectral conjecture of Bollobás and Nikiforov [J. Combin. Theory Ser. B 97 (2007)] asserts that if $G$ is a $K_{r+1}$-free graph with $m$ edges, then $\lambda_1^2(G) + \lambda_2^2(G) \le (1-\frac{1}{r})2m$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] confirmed the conjecture in the case $r=2$. Using this base case, they proved further that $\lambda (G)\le \sqrt{m-1}$ for every non-bipartite triangle-free graph $G$, with equality if and only if $m=5$ and $G=C_5$. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented an improvement by showing $\lambda (G) \le \beta (m)$, where $\beta(m)$ is the largest root of $Z(x):=x^3-x^2-(m-2)x+m-3$. The equality in Zhai--Shu's result holds only if $m$ is odd and $G$ is obtained from the complete bipartite graph $K_{2,\frac{m-1}{2}}$ by subdividing exactly one edge. Motivated by this observation, Zhai and Shu proposed a question to find a sharp bound when $m$ is even. We shall solve this question by using a different method and characterize three kinds of spectral extremal graphs over all triangle-free non-bipartite graphs with even size. Our proof technique is mainly based on applying Cauchy's interlacing theorem of eigenvalues of a graph, and with the aid of a triangle counting lemma in terms of both eigenvalues and the size of a graph.

Convolutional neural network intelligent diagnosis method using small samples based on SK-CAM

In order to solve the dependence of convolutional neural networks (CNN) on large samples of training data, an intelligent fault diagnosis method based on spectral kurtosis (SK) and attention mechanism is proposed. Firstly, the SK algorithm is used to obtain two-dimensional fast kurtosis graphs from vibration signals, and the two-dimensional fast spectral kurtosis graphs are converted into one-dimensional kurtosis time-domain samples, which are used as the input of CNN. Then the channel attention module (CAM) is added to CNN, and the weight is increased in the channel domain to eliminate the interference of invalid features. The accuracy of fault identification can reach 99.8 % by applying the proposed method on the fault diagnosis experiment of rolling bearings. Compared with the traditional deep learning (DL) method, the proposed method not only has higher accuracy, but also has lower dependence on the number of samples.

The spectral radius of H2k-free graphs

A fan, denoted by H2k, is the graph obtained by joining a vertex to a path with 2k vertices. A graph is H2k-free if it does not contain a subgraph H2k. Yuan (2021) [13] proved the Turán number and the extremal graphs of H2k-free graphs. In this paper, we characterize spectral extremal graphs of H2k-free graphs for sufficiently large n and k≥3.

PREDICTION OF ELECTROCHEMICAL PROCESSES OF LOCAL DISSOLUTION OF STAINLESS STEELS. PART 1. SPECTRAL –FREQUENCY ANALYSIS OF POTENTIAL FLUCTUATIONS

The propagation of localized corrosion of stainless steels was investigated by spectral frequency analysis (HPA) of potential fluctuations in the modes of stationary and cyclic galvanostatic polarization. Corrosion-resistant austenitic and austenitic-ferritic chromium-nickel steels were studied: 12X18N10T, 10X17N13M2T, 08X22N6T, resistant to pitting corrosion, in an aqueous solution of 0.1 M NaCl with a cyanide-containing complexing agent, a corrosion stimulator - 0.01 g/l (K3[Fe(CN)6]). The studies were carried out in the range of low densities (0.5–10 µA /cm2) due to the small values of the limiting diffusion current density in oxygen. To find the time of the origin of macropittings, it is proposed to use the NAV of chronopotentiograms with their subsequent statistical processing (Fourier transform). It has been established that fluctuations in the boundary polarization regime are characteristic of the process of the origin, development and passivation of macropittings. In the mode of stationary galvanostatic polarization, frequency characteristics were studied in the range of 1.0 ± 0.5 µA/cm2. When analyzing spectral density graphs, it was found that the dominant frequency characterizes the time of formation of the macropitting, and the sections of the chronopotentiogram should be divided into sections with a duration of no more than 50-100 s. In non-stationary conditions of galvanostatic polarization in the range of 1.0-10 µA/cm2 in a cyclic mode, it was found that the processes of nucleation and passivation of macropittings can be traced with a changing value of the polarizing current. Based on the frequency range of chronopotentiograms, it was found that the average macropitting lifetime is 250-300 s (frequency range 0.033 - 0.004 Hz). The dominant frequencies are determined to identify the characteristic dependencies of the envelopes along the upper (fhg) and lower (fhg) boundaries of the potential change, based on them, an algorithm is proposed for selecting the values of the parameters of the prediction mode of the formation of macropitting processes.

Rapid diagnosis of rheumatoid arthritis and ankylosing spondylitis based on Fourier transform infrared spectroscopy and deep learning

ObjectiveRheumatoid arthritis and Ankylosing spondylitis are two common autoimmune inflammatory rheumatic diseases that negatively affect activities of daily living and can lead to structural and functional disability, reduced quality of life. Here, this study utilized Fourier transform infrared (FTIR) spectroscopy on dried serum samples and achieved early diagnosis of rheumatoid arthritis and ankylosing spondylitis based on deep learning models. MethodA total of 243 dried serum samples were collected in this study, including 81 samples each from ankylosing spondylitis, rheumatoid arthritis, and healthy controls. Three multi-scale convolutional modules with different specifications were designed based on the multi-scale convolutional neural network (MSCNN) to effectively fuse the local features to enhance the generalization ability of the model. The FTIR was then combined with the MSCNN model to achieve a non-invasive, fast, and accurate diagnosis of ankylosing spondylitis, rheumatoid arthritis, and healthy controls. ResultsSpectral analysis shows that the curves and waveforms of the three spectral graphs are similar. The main differences are distributed in the spectral regions of 3300–3250 cm−1, 3000–2800 cm−1, 1750–1500 cm−1, and 1500–1300 cm−1, which represent: Amides, fatty acids, cholesterol, proteins with a carboxyl group, amide II, free amino acids, and polysaccharides. Four classification models, namely artificial neural network (ANN), convolutional neural network (CNN), improved AlexNet model, and multi-scale convolutional neural network (MSCNN) were established. Through comparison, it was found that the diagnostic AUC value of the MSCNN model was 0.99, and the accuracy rate was as high as 0.93, which was much higher than the other three models. ConclusionThe study demonstrated the superiority of MSCNN in distinguishing ankylosing spondylitis from rheumatoid arthritis and healthy controls. FTIR may become a rapid, sensitive, and non-invasive means of diagnosing rheumatism.

Spectral extremal graphs for the bowtie

Let Fk be the (friendship) graph obtained from k triangles by sharing a common vertex. The Fk-free graphs of order n which attain the maximal spectral radius was firstly characterized by Cioabă, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)], and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)] under the condition that n is sufficiently large. In this paper, we get rid of the condition on n being sufficiently large if k=2. The graph F2 is also known as the bowtie. We show that the unique n-vertex F2-free spectral extremal graph is the balanced complete bipartite graph adding an edge in the vertex part with smaller size if n≥7, and the condition n≥7 is tight. Our result is a spectral generalization of a theorem of Erdős, Füredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995)], which states that ex(n,F2)=⌊n2/4⌋+1. Moreover, we study the spectral extremal problem for Fk-free graphs with given number of edges. In particular, we show that the unique m-edge F2-free spectral extremal graph is the join of K2 with an independent set of m−12 vertices if m≥8, and the condition m≥8 is tight.

Zoology of non-Hermitian spectra and their graph topology

We uncover the very rich graph topology of generic bounded non-Hermitian spectra, distinct from the topology of conventional band invariants and complex spectral winding. The graph configuration of complex spectra are characterized by the algebraic structures of their corresponding energy dispersions, drawing new intimate links between combinatorial graph theory, algebraic geometry, and non-Hermitian band topology. Spectral graphs that are conformally related belong to the same equivalence class, and are characterized by emergent symmetries not necessarily present in the physical Hamiltonian. The simplest class encompasses well-known examples such as the Hatano-Nelson and non-Hermitian SSH models, while more sophisticated classes represent novel multicomponent models with interesting spectral graphs resembling stars, flowers, and insects. With recent rapid advancements in metamaterials, ultracold atomic lattices, and quantum circuits, it is now feasible to not only experimentally realize such esoteric spectra, but also investigate the non-Hermitian flat bands and anomalous responses straddling transitions between different spectral graph topologies.

An easy-to-build optical spectrometer for the Rayleigh scattering investigation of coffee solution

Spectroscopy is an essential technique required in numerous analytical fields, yet spectrometers are often expensive instruments. In this study, we present an easy-to-build and low-cost optical spectrometer. The spectrometer is constructed from readily accessible materials including a white LED as the light source, a collimating lens, a cuvette, a slit, a DVD as the diffraction grating, and a webcam. This device is designed to measure the visible light spectra of liquid samples, which could then be processed into transmittance, absorbance, and spectral intensity graphs. We took measurements from several absorbing and scattering samples to evaluate the designed spectrometer. We chose solutions of food colouring as an absorbing medium and instant coffee as a scattering medium. We showed that instant coffee is a very good candidate for Rayleigh scattering experiments, and that the relative concentration of coffee can be investigated by these experiments.

Advantages and Limitations of Quantum Routing

The Swap gate is a ubiquitous tool for moving information on quantum hardware, yet it can be considered a classical operation because it does not entangle product states. Genuinely quantum operations could outperform Swap for the task of permuting qubits within an architecture, which we call routing. We consider quantum routing in two models: (1) allowing arbitrary two-qubit unitaries, or (2) allowing Hamiltonians with norm-bounded interactions. We lower bound the circuit depth or time of quantum routing in terms of spectral properties of graphs representing the architecture interaction constraints, and give a generalized upper bound for all simple connected $n$-vertex graphs. In particular, we give conditions for a superpolynomial classical-quantum routing separation, which exclude graphs with a small spectral gap and graphs of bounded degree. Finally, we provide examples of a quadratic separation between gate-based and Hamiltonian routing models with a constant number of local ancillas per qubit and of an $\Omega(n)$ speedup if we also allow fast local interactions.

Comparative Study of Seismic analysis of Bridge Substructure in different Seismic Zones as per IRC Guidelines

Abstract: The capacity design philosophy has currently become design norm for the seismic design of structural systems. it is necessary to assess the overstrength capacity of piers before proceeding with the design of the foundation and superstructure. This paper is devoted to developing deterministic procedures for the seismic analysis of substructure and foundation. Therefore, a moment-curvature approach is analysed. A parametric study is then conducted to investigate the factors that causes the seismic forces in the system. A simplified analysis methodology is put forward based on IRC SP 114; 2018. It is applicable for seismic design of bridges with a design service life of 100 years, considering Design Basis Earthquake (DBE). It has covered the seismic map and spectral acceleration graphs as specified in IS: 1893-Part-I- 2016. It also adopts the method prescribed for evaluation of liquefaction possibility, as specified in IS: 1893-Part-I- 2016. For the evaluation of seismic forces, Elastic Seismic Acceleration method, Elastic Response Spectrum method and Linear Time History method are specified. The IRC Guidelines describe the various types of special investigations to be carried out for bridges to be constructed in near field zones, skew, and curved bridges and so on. For loads and load combinations, IRC 6-2017 provides the guidelines and specifications. Objective of this code is to provide common procedure for design of bridges. It deals with the various loads such as vehicular loads, braking forces, wind load, water current forces and their combinations. Keywords: Seismic design of Bridge Substructure, IRC guidelines, Seismic design, Seismic analysis, seismic zones.

Spectral extremal graphs for intersecting cliques

The (k,r)-fan is the graph consisting of k copies of the complete graph Kr which intersect in a single vertex, and is denoted by Fk,r. Erdős et al. (1995) [14] determined the maximum number of edges in an n-vertex graph that does not contain Fk,3 as a subgraph. Furthermore, Chen et al. (2003) [5] proved the analogous result on Fk,r for the general case r≥3. In this paper, we show that for sufficiently large n, the graphs of order n that contain no copy of Fk,r and attain the maximum spectral radius are also edge-extremal. That is, such graphs must have ex(n,Fk,r) edges.

Lattice fermions as spectral graphs

We study lattice fermions from the viewpoint of spectral graph theory (SGT). We find that a fermion defined on a certain lattice is identified as a spectral graph. SGT helps us investigate the number of zero eigenvalues of lattice Dirac operators even on the non-torus and non-regular lattice, leading to understanding of the number of fermion species (doublers) on lattices with arbitrary topologies. The procedure of application of SGT to lattice fermions is summarized as follows: (1) One investigates a spectral graph corresponding to a lattice fermion. (2) One obtains a matrix corresponding to the graph. (3) One finds zero eigenvalues of the matrix by use of the discrete Fourier transformation (DFT). (4) By taking an infinite-volume and continuum limits, one finds the number of species. We apply this procedure to the known lattice fermion formulations including Naive fermions, Wilson fermions and Domain-wall fermions, and reproduce the known fact on the number of species. We also apply it to the lattice fermion on the discretized four­dimensional hyperball and discuss the number of fermion species on the bulk. In the end of the paper, we discuss the application of the analysis to lattice fermions on generic lattices with arbitrary topologies, which could lead to constructing a new theorem regarding the number of species.

Validating Poly(3,4-ethylene dioxythiophene) Polystyrene Sulfonate-Based Textile Electroencephalography Electrodes by a Textile-Based Head Phantom.

It is important to go through a validation process when developing new electroencephalography (EEG) electrodes, but it is impossible to keep the human mind constant, making the process difficult. It is also very difficult to identify noise and signals as the input signal is unknown. In this work, we have validated textile-based EEG electrodes constructed from a poly(3,4-ethylene dioxythiophene) polystyrene sulfonate:/polydimethylsiloxane coated cotton fabric using a textile-based head phantom. The performance of the textile-based electrode has also been compared against a commercial dry electrode. The textile electrodes collected a signal to a smaller skin-to-electrode impedance (−18.9%) and a higher signal-to-noise ratio (+3.45%) than Ag/AgCl dry electrodes. From an EEGLAB, it was observed that the inter-trial coherence and event-related spectral perturbation graphs of the textile-based electrodes were identical to the Ag/AgCl electrodes. Thus, these textile-based electrodes can be a potential alternative to monitor brain activity.

The spectral radius of graphs with no odd wheels

The odd wheel W2k+1 is the graph formed by joining a vertex to a cycle of length 2k. In this paper, we investigate the largest value of the spectral radius of the adjacency matrix of an n-vertex graph that does not contain W2k+1. We determine the structure of the spectral extremal graphs for all k≥2,k⁄∈{4,5}. When k=2, we show that these spectral extremal graphs are among the Turán-extremal graphs on n vertices that do not contain W2k+1 and have the maximum number of edges, but when k≥9, we show that the family of spectral extremal graphs and the family of Turán-extremal graphs are disjoint.

Dual Sparse Representation Graph-Based Copropagation for Semisupervised Hyperspectral Image Classification

Graph-based semisupervised hyperspectral image (HSI) classification methods have obtained extensive attention. In graph-based methods, a graph is first constructed, and then the label propagation is carried out on the constructed graph to obtain the labels for unknown samples. However, the results of label propagation may be unreliable, especially in the case of very limited labeled samples. To address the above problem, we propose dual sparse representation graph-based collaborative propagation (DSRG-CP) for HSI classification. Specifically, DSRG-CP adopts sparse representation (SR) to construct spectral and spatial graphs on spectral and spatial dimensions, respectively. Then, label propagation is performed on two graphs iteratively. In each iteration, only the samples with high classification confidence from one graph are added into another graph as labeled samples for the next label propagation. After several iterations, the labels of unlabeled samples are predicted by fusing the results of label propagation from two graphs. In addition, to make the spectral graph more discriminative, the regularizer of spectral statistical information is added into spectral SR model. To make the classification results more consistent in space, the superpixel block constraint is added into spatial graph model as regularizer. To evaluate the performance of the proposed method, DSRG-CP is compared with several graph-based methods and other state-of-the-art methods. Extensive experiments on real HSI data sets show that DSRG-CP can obtain competitive results for HSI classification.