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In this paper, we mainly concern the fixed-time consensus control problem for nonlinear leader-following multi-agent systems(MASs). An edge-based adaptive control protocol is presented with part of parameters updated by relative information of system state. Compared with the common node-based adaptive protocols, the proposed edge-based adaptive control protocol assigning different time-varying weights to different communication edges is more accurate. By selecting special Lyapunov functions and theoretical analysis, the obtained conclusions indicate that both sufficient conditions and the settling time of the fixed-time consensus are independent of the system's global information such as the spectrum of Laplacian matrix, which implies that the given protocol is fully distributed. Finally, numerical simulation is illustrated to support the correctness of the main theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we prove an existence result for a quasilinear elliptic variational inequality of the form u ∈ K ⊂ V : 0 ∈ - Δpu+aF(u) + ...IK(u) ∈ V* in the whole ℝ N under bilateral constraints K given by K = {v ∈ V : ϕ (x) ≤ v(x) ≤ Ψ(x) a.e. in ℝ N}, where Δp is the p-Laplacian, the underlying solution space V is the homogeneous Sobolev space (also called Beppo-Levi space) V = D1,p(ℝN) with 1 < p < N, and IK : V → ℝ U {+∞} is the indicator functional corresponding to K with its subdifferential ...IK. The lower order Nemytskij operator F is generated by a Carathéodory function f : ℝN x ℝ → ℝ, and the measurable and bounded coefficient α is supposed to decay like |x|- (N+α) at infinity. The growth conditions that we impose on f are such that the operator -- Δp + aF : V → V*, in general, is not coercive with respect to K which prevents us from applying standard existence results. Another difficulty, which arises due to the lack of compact embedding of V into Lq(ℝN) spaces, needs to be overcome in an appropriate way. Without assuming additional assumptions such as the existence of sub- and supersolutions, we are able not only to prove the existence of solutions, but also show the compactness of the set of all solutions in V. Finally, an extension of the theory is established, which allows us to deal with noncoercive bilateral variational-hemivariational inequalities in ℝN. The proof of our main existence result is based on a modified penalty approach. [ABSTRACT FROM AUTHOR]

Let A = (aij) and H = (hij) be positive semidefinite matrices of the same order. If aij ≥ |hij| for all i, j; A is diagonally dominant and all row sums of H are equal to zero, then we show that the sum of all k x k principal minors of A is greater than or equal to the sum of all k x k principal minors of H. [ABSTRACT FROM AUTHOR]

In this paper, the unique graph with the minimal second smallest normalized Laplacian eigenvalue of unicyclic graphs with fixed girth is determined. [ABSTRACT FROM AUTHOR]

Hyper-networks tend to perform better in representing multivariate relationships among nodes. Yet, due to the complexity of the hyper-network structure, research in synchronization dynamics is rarely involved. In this paper, a Kuramoto model more suitable for k-uniform hyper-networks is proposed. And the generalized Laplacian matrix expression of the k-uniform hyper-network is present. We use the eigenvalue ratio of the generalized Laplacian matrix to quantify synchronization. And we studied the effects of some important structure parameters on the synchronization of three types of k-uniform hyper-networks. And obtained different relationships between synchronization and these parameters. The results show the synchronization of the k-uniform hyper-networks is related to both structure and parameters. And as the size of the nodes increases, the synchronization ability gradually increases for ER random hyper-network, while that gradually decreases for NW small-world hyper-network and BA scale-free hyper-network. As the uniformity increases, the synchronization ability of all three types of uniform hyper-networks increases. In addition, when the structure and node size are fixed, the synchronization ability increases with the increase of the hyper-clustering coefficient in BA scale-free hyper-network and ER random hyper-network, while it decreases with the increase of the hyper-clustering coefficient in NW small-world hyper-network. [ABSTRACT FROM AUTHOR]

We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we give a combinatorial interpretation for the coefficients of the weighted Laplacian characteristic polynomial in terms of weighted spanning forests and use this to prove a deletion-contraction relation. We prove various interlacing theorems relating to deletion and contraction, as well as to rectangular tilings, drawing on the work of Brooks, Smith, Stone and Tutte on square tilings. Additionally, we show that the weighted Laplacian also satisfies a vertex analogue of deletion-contraction. We give applications of weighted Laplacian eigenvalues to sparse cuts, independent sets and graph colouring, and establish new cases of a problem of Stanley on distinguishing nonisomorphic trees. [ABSTRACT FROM AUTHOR]

New SPICE-compatible circuit models of a multiport are presented here that are suitable for the frequency-domain and time-domain analyses of hybrid systems containing linear distributed elements and possibly non-linear lumped elements. Distributed elements models are based on scattering parameters with potentially complex reference impedances, which are not necessarily equal for all ports. Both exact and approximated (lumped) models are proposed. The scattering parameters are directly taken as the model element values in the former case. In the latter case, the model element values are equal to the real and imaginary parts of the poles and residues of the rational approximation. The models comprise a multiport (with an admittance matrix numerically equal to the modeled scattering matrix or approximating it) equipped with a pair of coupled impedances at each port. A few examples validate the proposed approach and prove its efficiency in terms of matrix size and analysis time compared to some selected commercial counterparts. [ABSTRACT FROM AUTHOR]

Let S(Γ) be a Seidel matrix of a graph Γ of order n and let D(Γ) = diag(n − 1 − 2d1, n − 1 − 2d2, ..., n − 1 − 2dn) be a diagonal matrix with di denoting the degree of a vertex vi in Γ. The Seidel Laplacian matrix of Γ is defined as SL(Γ) = D(Γ) − S(Γ). In this paper, we obtain an upper bound, and a lower bound on the Seidel Laplacian Estrada index of graphs. Moreover, we find a relation between Seidel energy and Seidel Laplacian energy of graphs. We establish some lower bounds on the Seidel Laplacian energy in terms of different graph parameters. Finally, we present a relation between Seidel Laplacian Estrada index and Seidel Laplacian energy of graphs. [ABSTRACT FROM AUTHOR]

We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which corresponds to a specific clustered state. The network's nodes involved in each of the clusters can be identified, and the value of the coupling strength at which the events are taking place can be approximately ascertained. Finally, we present large-scale simulations which show the accuracy of the approximation made, and of our predictions in describing the synchronization transition of both synthetic and real-world large size networks, and we even report that the observed sequence of clusters is preserved in heterogeneous networks made of slightly non-identical systems. Natural and synthetic networked systems can be characterized by synchrony or asynchrony of groups of their nodes (clusters) and undergo different scenarios of synchronization transitions. The authors propose an approach to predict the entire sequence of events that are taking place during the synchronization transition in networks of identical elements with any preferrable architecture. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n, where n is a positive integer. A spanning subgraph F of G is called a path-factor if every component of F is a path of order at least 2. A P ≥ k -factor means a path-factor in which every component admits order at least k ( k ≥ 2 ). The distance matrix D (G) of G is an n × n real symmetric matrix whose (i, j)-entry is the distance between the vertices v i and v j . The distance signless Laplacian matrix Q (G) of G is defined by Q (G) = T r (G) + D (G) , where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue η 1 (G) of Q (G) is called the distance signless Laplacian spectral radius of G. In this paper, we aim to present a distance signless Laplacian spectral radius condition to guarantee the existence of a P ≥ 2 -factor in a graph and claim that the following statements are true: (i) G admits a P ≥ 2 -factor for n ≥ 4 and n ≠ 7 if η 1 (G) < θ (n) , where θ (n) is the largest root of the equation x 3 - (5 n - 3) x 2 + (8 n 2 - 23 n + 48) x - 4 n 3 + 22 n 2 - 74 n + 80 = 0 ; (ii) G admits a P ≥ 2 -factor for n = 7 if η 1 (G) < 25 + 161 2 . [ABSTRACT FROM AUTHOR]

Laplacian matrix and its spectrum are commonly used for giving a measure in networks in order to analyse its topological properties. In this paper, Laplacian matrix of graphs and their spectrum are studied. Laplacian energy of a graph G of order n is defined as LE (G) = ∑ i = 1 n | λ i (L) - d ¯ | , where λ i (L) is the i-th eigenvalue of Laplacian matrix of G, and d ¯ is their average. If LE (G) = LE (K n) for the complete graph K n of order n, then G is known as L-borderenergetic graph. In the first part of this paper, we construct three infinite families of non-complete disconnected L-borderenergetic graphs: Λ 1 = { G b , j , k = [ (((j - 2) k - 2 j + 2) b + 1) K (j - 1) k - (j - 2) ] ∪ b (K j × K k) | b , j , k ∈ Z + } , Λ 2 = { G 2 , b = [ K 6 ∇ b (K 2 × K 3) ] ∪ (4 b - 2) K 9 | b ∈ Z + } , Λ 3 = { G 3 , b = [ b K 8 ∇ b (K 2 × K 4) ] ∪ (14 b - 4) K 8 b + 6 | b ∈ Z + } , where ∇ is join operator and × is direct product operator on graphs. Then, in the second part of this work, we construct new infinite families of non-complete connected L-borderenergetic graphs Ω 1 = { K 2 ∇ a K 2 r ¯ | a ∈ Z + } , Ω 2 = { a K 3 ∪ 2 (K 2 × K 3) ¯ | a ∈ Z + } and Ω 3 = { a K 5 ∪ (K 3 × K 3) ¯ | a ∈ Z + } , where G ¯ is the complement operator on G. [ABSTRACT FROM AUTHOR]

This work investigates the emergence of structural super-diffusion in finite multiplex networks, focusing on situations where long-range interactions (LRIs) are present in at least one of the layers. Employing the Lévy random walk model, we explore how the likelihood of observing LRIs and the strength of the coupling among layers, specifically through the inter-layer diffusion coefficient, affect the relaxation time of the entire multiplex network. Our aim is to determine if this collective relaxation time is shorter compared to that in isolated layer configurations. We quantify the relaxation times through algebraic connectivity, the second-smallest eigenvalue of the network's (Supra-)Laplacian matrix. The formalism is adapted to scenarios where long-range jumps may exist by considering Mellin or Laplace transforms. We study samples of multiplexes whose layers are obtained from the models by Erdös-Rény, Watts-Strogatz and Barabási-Albert, and discuss in great detail the results for multiplexes with two layers. Besides the rather common enhancement of multiplex diffusion in comparison to the diffusion observed in their isolated layers, our results show that the timescale of layers with LRIs may be still faster than that due only to the usual multiplex network, speeding-up the whole system's diffusion. In addition, we also provide enough evidences that a novel linear diffusion regime emerges with strong inter-layer coupling and the presence of LRIs in one or more layers. Graphical Abstract highlights the main focus of the work, the occurrence of superdiffusion in a multiplex with M = 2 layers. The connectivity ratio η = Λ 2 / max (λ 2 1 , λ 2 2) (red line) crosses the dashed line η = 1 at a well defined value of D x , and remains in the super-diffusion region highlighted in green [ABSTRACT FROM AUTHOR]

This paper considers the secure distributed control consensus problem of linear multi‐agent systems (MASs) under switching topologies, subject to intermittent sequential scaling attacks, which was compelled to scaling factor, attack frequency, duration and cooperative‐competitive networks. First, the scenario of a fixed topology is considered, and a novel control protocol combined with a logarithmic quantizer and relative state measurements of neighbouring agents is discussed. The sighed graph is utilized to characterize the communication topology determined by the information flow directions and captured by the graph Laplacian matrix. After that, sufficient conditions for effectiveness of the developed control methods in guiding the MASs to secure bipartite leader‐following consensus are constructed. Second, the scenario of switching topologies is considered, and it is derived that the secure bipartite consensus will be achieved if the designed state feedback control protocol with the scaling factor, the attack duration, attack frequency and switching signal are selected properly. At last, to prove the effectiveness of the designed controllers, a simulation example depended on the real‐word actual military is introduced. [ABSTRACT FROM AUTHOR]

Graph energy is a measurement of determining the structural information content of graphs. The first Zagreb index can be handled with its connection to graph energy. In this paper, a novel and significant application of the first Zagreb index to composite graphs based on fractal graphs is revealed, and by the relation between quasi-Laplacian energy and the vertex degrees of a graph, we derive closed-form formulas for the quasi-Laplacian energy of fractal graphs or namely R-graphs, R-vertex and edge join, R-vertex and edge corona, R-vertex and edge neighborhood graphs in terms of the corresponding energy, the first Zagreb indices, number of vertices and edges of the underlying graphs. [ABSTRACT FROM AUTHOR]

This paper proposes a new design method of a linear-to-circular polarization converter based on the branch line circuit model. The series branches are implemented by dielectric spacers and the shunt branches are implemented by anisotropic admittance sheets. The admittance matrix of the anisotropic admittance sheets is derived by even and odd mode analysis. A unit cell consisting of two metal layers and a dielectric spacer is then designed using an iterative algorithm for operation from 15.5 to 23 GHz. Results indicate that the axial ratio and insertion loss of the designed polarizer are better than 3 and 1.5 dB respectively, for incident angles between ±45°. To validate the simulation results, a waveguide-based test setup is designed and fabricated, with measurement results indicating good agreement with simulations. The proposed method can be potentially applied to design other anisotropic structures such as half-plate polarizers and polarization rotators. [ABSTRACT FROM AUTHOR]

The low-frequency oscillation (LFO) has occurred in the train–network system due to the introduction of the power electronics of the trains. The modeling and analyzing method in current researches based on electrified railway unilateral power supply system are not suitable for the LFO analysis in a bilateral power supply system, where the trains are supplied by two traction substations. In this work, based on the single-input and single-output impedance model of China CRH5 trains, the node admittance matrices of the train–network system both in unilateral and bilateral power supply modes are established, including three-phase power grid, traction transformers and traction network. Then the modal analysis is used to study the oscillation modes and propagation characteristics of the unilateral and bilateral power supply systems. Moreover, the influence of the equivalent inductance of the power grid, the length of the transmission line, and the length of the traction network are analyzed on the critical oscillation mode of the bilateral power supply system. Finally, the theoretical analysis results are verified by the time-domain simulation model in MATLAB/Simulink. [ABSTRACT FROM AUTHOR]

This paper delves into three research directions, leveraging the Lovász ϑ-function of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses of strongly regular graphs, and extending prior results. The second part explores cospectral and nonisomorphic graphs, drawing on a work by Berman and Hamud (2024), and it derives related properties of two types of joins of graphs. For every even integer such that n≥14, it is constructively proven that there exist connected, irregular, cospectral, and nonisomorphic graphs on n vertices, being jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, while also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lovász ϑ-functions. The third part focuses on establishing bounds on graph invariants, particularly emphasizing strongly regular graphs and triangle-free graphs, and compares the tightness of these bounds to existing ones. The paper derives spectral upper and lower bounds on the vector and strict vector chromatic numbers of regular graphs, providing sufficient conditions for the attainability of these bounds. Exact closed-form expressions for the vector and strict vector chromatic numbers are derived for all strongly regular graphs and for all graphs that are vertex- and edge-transitive, demonstrating that these two types of chromatic numbers coincide for every such graph. This work resolves a query regarding the variant of the ϑ-function by Schrijver and the identical function by McEliece et al. (1978). It shows, by a counterexample, that the ϑ-function variant by Schrijver does not possess the property of the Lovász ϑ-function of forming an upper bound on the Shannon capacity of a graph. This research paper also serves as a tutorial of mutual interest in zero-error information theory and algebraic graph theory. [ABSTRACT FROM AUTHOR]

ChatGPT represents a significant milestone in the field of artificial intelligence (AI), finding widespread applications across diverse domains. However, its effectiveness in mathematical contexts has been somewhat constrained by its susceptibility to conceptual errors. Concurrently, topological data analysis (TDA), a relatively new discipline, has garnered substantial interest in recent years. Nonetheless, the advancement of TDA is impeded by the limited understanding of computational algorithms and coding proficiency among theoreticians. This work endeavors to bridge the gap between theoretical topological concepts and their practical implementation in computational topology through the utilization of ChatGPT. We showcase how a pure theoretician, devoid of computational experience and coding skills, can effectively transform mathematical formulations and concepts into functional codes for computational topology with the assistance of ChatGPT. Our strategy outlines a productive process wherein a mathematician trains ChatGPT on pure mathematical concepts, steers ChatGPT towards generating computational topology codes, and subsequently validates the generated codes using established examples. Our specific case studies encompass the computation of Betti numbers, Laplacian matrices, and Dirac matrices for simplicial complexes, as well as the persistence of various homologies and Laplacians. Furthermore, we explore the application of ChatGPT in computing recently developed topological theories for hypergraphs and digraphs, as well as the persistent harmonic space, which has not been computed in the literature, to the best of our knowledge. This work serves as an initial step towards effectively transforming pure mathematical theories into practical computational tools, with the ultimate goal of enabling real applications across diverse fields. [ABSTRACT FROM AUTHOR]

Predicting stock trends in financial markets is of significant importance to investors and portfolio managers. In addition to a stock's historical price information, the correlation between that stock and others can also provide valuable information for forecasting future returns. Existing methods often fall short of straightforward and effective capture of the intricate interdependencies between stocks. In this research, we introduce the concept of a Laplacian correlation graph (LOG), designed to explicitly model the correlations in stock price changes as the edges of a graph. After constructing the LOG, we will build a machine learning model, such as a graph attention network (GAT), and incorporate the LOG into the loss term. This innovative loss term is designed to empower the neural network to learn and leverage price correlations among different stocks in a straightforward but effective manner. The advantage of a Laplacian matrix is that matrix operation form is more suitable for current machine learning frameworks, thus achieving high computational efficiency and simpler model representation. Experimental results demonstrate improvements across multiple evaluation metrics using our LOG. Incorporating our LOG into five base machine learning models consistently enhances their predictive performance. Furthermore, backtesting results reveal superior returns and information ratios, underscoring the practical implications of our approach for real-world investment decisions. Our study addresses the limitations of existing methods that miss the correlation between stocks or fail to model correlation in a simple and effective way, and the proposed LOG emerges as a promising tool for stock returns prediction, offering enhanced predictive accuracy and improved investment outcomes. [ABSTRACT FROM AUTHOR]

Let R be the commutative ring R = Zp²[x]/〈x²〉 with identity and Z*(R) be the set of all non-zero zero-divisors of R. Then, (R) is said to be a zero-divisor graph if and only if a · b = 0 where a; b 2 V (Γ(R)) = Z*(R) and (a; b) 2 E(Γ(R)). Let λ1, λ2,... λ2 be the eigenvalues of the adjacency matrix, and let µ1, µ2,..., µn be the eigenvalues of the Laplacian matrix of Γ(R). Then we discuss the energy E(Γ(R)) = Pn i=1jij and the Laplacian energy LE(Γ(R)) = Pn i=1 λi Γ 2m n where n and m are the order and size of Γ(R). [ABSTRACT FROM AUTHOR]

The Laplacian eigenvalues and polynomials of the networks play an essential role in understanding the relations between the topology and the dynamic of networks. Generally, computation of the Laplacian spectrum of a network is a hard problem and there are just a few classes of graphs with the property that their spectra have been completely computed. Laplacian spectrum for n-prism networks was investigated in [Liu et al., Neurocomputing 198 (2016) 69-73]. In this paper, we give a method for calculating the eigenvalues and characteristic polynomial of the Laplacian matrix of a generalized n-prism network. We show how such large networks can be constructed from small graphs by using graph products. Moreover, our results are used to obtain the Kirchhoff index and the number of the spanning trees in the generalized n-prism networks. We also give some examples of applications, that explain the usefulness and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Consider a simple, undirected graph G = (V;E), where A represents the adjacency matrix and Q represents the Laplacian matrix of G. The second smallest eigenvalue of Laplacian matrix of G is called the algebraic connectivity of G. In this article, we present a Python program for studying the Laplacian eigenvalues of a graph. Then, we determine the unique graph of minimum algebraic connectivity in the set of all tricyclic graphs. [ABSTRACT FROM AUTHOR]

Recently, graph theory has become a promising tool for biomedical signal analysis, wherein the signals are transformed into a graph network and represented as either adjacency or Laplacian matrices. However, as the size of the time series increases, the dimensions of transformed matrices also expand, leading to a significant rise in computational demand for analysis. Therefore, there is a critical need for efficient feature extraction methods demanding low computational time. This paper introduces a new feature extraction technique based on the Gershgorin Circle theorem applied to biomedical signals, termed Gershgorin Circle Feature Extraction (GCFE). The study makes use of two publicly available datasets: one including synthetic neural recordings, and the other consisting of EEG seizure data. In addition, the efficacy of GCFE is compared with two distinct visibility graphs and tested against seven other feature extraction methods. In the GCFE method, the features are extracted from a special modified weighted Laplacian matrix from the visibility graphs. This method was applied to classify three different types of neural spikes from one dataset, and to distinguish between seizure and non-seizure events in another. The application of GCFE resulted in superior performance when compared to seven other algorithms, achieving a positive average accuracy difference of 2.67% across all experimental datasets. This indicates that GCFE consistently outperformed the other methods in terms of accuracy. Furthermore, the GCFE method was more computationally-efficient than the other feature extraction techniques. The GCFE method can also be employed in real-time biomedical signal classification where the visibility graphs are utilized such as EKG signal classification. [ABSTRACT FROM AUTHOR]

In this article, the distributed adaptive asymptotical consensus tracking control is considered for a class of high‐order nonlinear multi‐agent systems with matched unknown parameters under the event‐triggered communication mechanism. The controllers, adaptive laws and triggering conditions are presented under directed topology with a spanning tree, relieving communication burden among the connected agents since continuous monitoring of in‐neighbors is no longer needed. To handle the cross‐coupling term associated with asymmetric Laplacian matrix and unknown parameters of all agents when the candidate Lyapunov function is defined in terms of local consensus errors, adaptive immersion and invariance control technique is adopted. Adaptive estimators are also designed to estimate the constants related to global information, with which the proposed control scheme is fully distributed. It is shown that all closed‐loop system signals are globally uniformly bounded and all agents can track the reference signal asymptotically while ruling out Zeno behavior though a subset of the agents cannot have direct access to the reference signal information. A numerical example is illustrated to show the effectiveness of the proposed control protocol. [ABSTRACT FROM AUTHOR]

Graph clustering aims to divide nodes into different clusters without labels and has attracted great attention due to the success of graph neural networks (GNNs). Traditional GNN-based clustering methods are based on the homophilic assumption, i.e., connected nodes belong to the same clusters. However, this assumption is not always true, as heterophilic graphs are also ubiquitous in the real world, which limits the application of GNNs. Furthermore, these methods overlook global positions, which can result in erroneous clustering. To solve the aforementioned problems, we propose a novel model called Preserving Global Information for Graph Clustering with Masked Autoencoders (GCMA). We first propose a low–high-pass filter to capture meaningful low- and high-frequency information. Then, we propose a graph diffusion method to obtain the global position. Specifically, a parameterized Laplacian matrix is proposed to better control the global direction. To further enhance the learning ability of the autoencoders, we design a model with a masking strategy that enhances the learning ability. Extensive experiments on both homophilic and heterophilic graphs demonstrate GCMA's advantages over state-of-the-art baselines. [ABSTRACT FROM AUTHOR]

With the increased complexity of power systems stemming from the connection of high-proportion renewable energy sources, coupled with the escalating volatility and uncertainty, the key transmission sections that serve as indicators of the power grid's security status are also subject to frequent changes, posing challenges to grid monitoring. The search method for key transmission sections based on an improved spectral clustering algorithm is proposed in this paper. A branch weight model, considering the impact of node voltage and power flow factors, is initially established to comprehensively reflect the electrical connectivity between nodes. Subsequently, a weighted graph model is constructed based on spectral graph theory, and an improved spectral clustering algorithm is employed to partition the power grid. Finally, a safety risk indicator is utilized to identify whether the partitioned sections are key transmission sections. Results from case studies on the IEEE39-node system and actual power grid examples demonstrate that the proposed method accurately and effectively searches for all key transmission sections of the system and identifies their security risks. The application in real power grid scenarios validates its ability to screen out some previously unrecognized key transmission sections. [ABSTRACT FROM AUTHOR]

The equivalent electro-mechanical admittance matrix of cylindrical piezo-electrics with axial polarization was derived, and based on the electrical terms of this matrix, the resonant and anti-resonant frequencies of the cylindrical piezo-electrics were obtained. Then, these results were compared with the obtained experimental values of resonant and anti-resonant frequencies. The obtained results show an average erorr 2 % for resonant and 5 % for anti-resonant frequencies, which have good precision and matching. In this method, the cylindrical piezo-electric was modeled by using the force-velocity boundary conditions and equalizing the mechanical elements with the electrical ones. The force-velocity equalizing provides the anti-resonant frequency of the unit, which is required to design the piezo-electric transducers for sensing applications. This method is useful for modeling piezo-electric transducers composed of several stacks of cylindrical shaped piezo-electrics in both longitudinal and flexural stimulation types. The expression in matrix notation provides a simple way for modeling and analyzing multilayered piezo-electric transducers compared to the analytical method and the finite element analysis with more pricision and lower time and cost-consumption, respectively. [ABSTRACT FROM AUTHOR]

This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in if(div). The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation--histopolation basis (cf. [W. Pazner, T. Kolev, and C. R. Dohrmann, SIAM J. Sci. Comput., 45 (2023), pp. A675-A702]), efficient matrix-free preconditioners can be constructed for the (1,1)-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the "crooked pipe" grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case. [ABSTRACT FROM AUTHOR]

Amidst the rapid advancements in technology, there is a growing demand for processing an increasing volume and quality of images, which necessitates faster image processing capabilities. Enhancing the efficiency of image processing algorithms has thus become a critical priority. Existing quantum image edge detection algorithms tend to exhibit high circuit complexity, which is directly linked to the dimensions of the images being processed, leading to less than optimal computational velocities. In this study, we introduce a quantum image edge detection algorithm that is based on the Laplacian of Gaussian operator. This novel algorithm capitalizes on the quantum parallelism of quantum computing, resulting in a marked enhancement in both the speed and performance of edge detection. To substantiate the practicality of our approach, we conduct simulations using the International Business Machines Quantum (IBM Q) platform. The circuit complexity of our algorithm is meticulously computed, revealing a lower complexity compared to analogous quantum edge detection algorithms. Notably, this complexity is detached from the image size and is solely contingent upon the grayscale value range of the image pixels. [ABSTRACT FROM AUTHOR]

Ever concluded whereas fractal fluctuations exhibit quantum-like chaos. Then provided a configurative description of a repeating units [from gather (unit) to transform (emergent unit)] of four conservation laws& forces in fractal organization of nature. To be proofed that Fibonacci sequence performed through Laplacian matrix normalization which technoeconophysics realm assumed as financial innovation. [ABSTRACT FROM AUTHOR]

The use of the global mean first-passage time (GMFPT) in random walks on networks has been widely explored in the field of statistical physics, both in theory and practical applications. The GMFPT is the estimated interval of time needed to reach a state j in a system from a starting state i. In contrast, there exists an intrinsic measure for a stochastic process, known as Kemeny's constant, which is independent of the initial state. In the literature, it has been used as a measure of network efficiency. This article deals with a graph-spectrum-based method for finding both the GMFPT and Kemeny's constant of random walks on spiro-ring networks (that are organic compounds with a particular graph structure). Furthermore, we calculate the Laplacian matrix for some specific spiro-ring networks using the decomposition theorem of Laplacian polynomials. Moreover, using the coefficients and roots of the resulting matrices, we establish some formulae for both GMFPT and Kemeny's constant in these spiro-ring networks. [ABSTRACT FROM AUTHOR]

For a simple signed graph G with the adjacency matrix A and net degree matrix $ D^{\pm } $ D ± , the net Laplacian matrix is $ L^{\pm }=D^{\pm }-A $ L ± = D ± − A. We introduce a new oriented incidence matrix $ N^{\pm } $ N ± which can keep track of the sign as well as the orientation of each edge of G. Also $ L^{\pm }=N^{\pm }(N^{\pm })^T $ L ± = N ± (N ± ) T . Using this decomposition, we find the number of both positive and negative spanning trees of G in terms of the principal minors of $ L^{\pm } $ L ± generalizing the Matrix Tree Theorem for an unsigned graph. We present similar results for the signless net Laplacian matrix $ Q^{\pm }=D^{\pm }+A $ Q ± = D ± + A along with a combinatorial formula for its determinant. [ABSTRACT FROM AUTHOR]

A signed network Σ = (G , σ) with the underlying graph G = (V , E) , used as a mathematical model for analyzing social networks, has each edge in E with a weight 1 or − 1 assigned by the signature function σ. In this paper, we deal with two types of Detour Distance Laplacian (DDL) matrices for signed networks. We characterize balance in signed social networks using these matrices and we compute the DDL spectrum of certain unbalanced signed networks, as balanced signed networks behave like unsigned ones. [ABSTRACT FROM AUTHOR]

In this paper, the secure containment control issue for heterogeneous multi-agent systems subject to hybrid attacks is studied. In the network channels among agents, adversaries launch replay attacks such that the followers can only retrieve the previous information of their neighbors. To characterize the effects of replay attacks, a distributed auxiliary system with heterogeneous time-varying delay is constructed to estimate the convex hull of the leaders' state. The control input of each follower may be contaminated by the malicious false data injection (FDI) attacks with unknown bounded time-varying signals, which are intended to compromise the containment performance of the whole system. A local adaptive compensator is designed for each follower to eliminate the adverse effects of the FDI attacks. Based on the distributed auxiliary system and the local adaptive compensator, a secure distributed control protocol is proposed to ensure that the output trajectories of each follower can converge to the dynamic convex hull spanned by the leader outputs. Finally, two illustrative examples are provided to show the feasibility of the proposed secure control scheme. • This paper introduces a secure control protocol to dispose a more complicated attack situation. • The original Laplacian matrix is divided into several sub-Laplacian matrices associated with the heterogeneous time-varying delays. • The designed adaptive compensator can fully eliminate the adverse effects of the FDI attacks without using any information of the FDI attacks. [ABSTRACT FROM AUTHOR]

The existing impulsive consensus algorithms for second-order Lipschitz nonlinear multi-agent systems require to apply the impulsive control to both position and velocity vectors at the same time. Such a requirement cannot be met in most of the real-world applications. To overcome the limitations of these impulsive algorithms, two kinds of new second-order impulsive consensus algorithms using only velocity regulation are proposed. Through developing a weighted discontinuous Lyapunov function-based approach that is able to leverage the spectral property of Laplacian matrix, impulse-dwell-time-dependent sufficient conditions for solving second-order impulsive consensus are derived in the form of linear matrix inequalities. Further, it is shown that if the impulsively controlled velocity subsystems are globally exponentially stable, the impulsive static consensus algorithm is able to ensure that all agents tend to an agreed position. Based on the consensus conditions, two convex optimization problems are formulated, by which the impulsive gain matrices for ensuring a prescribed exponential convergence rate can be designed. Finally, the effectiveness of the proposed distributed impulsive consensus algorithms is certified through numerical simulations. • Impulsive consensus algorithms based on velocity regulation are proposed for vector second-order Lipschitz nonlinear MASs. • The relationship between impulsive dynamical consensus and impulsive static consensus is analyzed. • A convex framework is proposed to design impulsive consensus gain matrices with guaranteed convergence rate. [ABSTRACT FROM AUTHOR]

For sparse subspace clustering methods, it is crucial to develop a good representation matrix to capture the data structure. In this paper, we incorporated the label information into sparse representation and proposed a new semi-supervised sparse subspace clustering method, named semi-supervised sparse subspace clustering with manifold regularization (S 4 CMR). When developing the sparse self-expressive matrix, the S 4 CMR method utilized the label information to constrain the development of expressiveness coefficients. The local manifold regularization was also integrated to enhance clustering stability and local consistency. By utilizing the Alternating Direction Method of Multipliers (ADMM), the convex optimization problem associated with linear constraints can be easily resolved. The developed similarity matrix can provide strong discriminant information, making it more effective for semi-supervised tasks. The effectiveness of the proposed algorithm is demonstrated through experiments on benchmark data sets, such as motion segmentation and image clustering. [ABSTRACT FROM AUTHOR]

In graph theory, the weighted Laplacian matrix is the most utilized technique to interpret the local and global properties of a complex graph structure within computer vision applications. However, with increasing graph nodes, the Laplacian matrix's dimensionality also increases accordingly. Therefore, there is always the "curse of dimensionality"; In response to this challenge, this paper introduces a new approach to reducing the dimensionality of the weighted Laplacian matrix by utilizing the Gershgorin circle theorem by transforming the weighted Laplacian matrix into a strictly diagonal domain and then estimating rough eigenvalue inclusion of a matrix. The estimated inclusions are represented as reduced features, termed GC features; The proposed Gershgorin circle feature extraction (GCFE) method was evaluated using three publicly accessible computer vision datasets, varying image patch sizes, and three different graph types. The GCFE method was compared with eight distinct studies. The GCFE demonstrated a notable positive Z-score compared to other feature extraction methods such as I-PCA, kernel PCA, and spectral embedding. Specifically, it achieved an average Z-score of 6.953 with the 2D grid graph type and 4.473 with the pairwise graph type, particularly on the E_Balanced dataset. Furthermore, it was observed that while the accuracy of most major feature extraction methods declined with smaller image patch sizes, the GCFE maintained consistent accuracy across all tested image patch sizes. When the GCFE method was applied to the E_MNSIT dataset using the K-NN graph type, the GCFE method confirmed its consistent accuracy performance, evidenced by a low standard deviation (SD) of 0.305. This performance was notably lower compared to other methods like Isomap, which had an SD of 1.665, and LLE, which had an SD of 1.325; The GCFE outperformed most feature extraction methods in terms of classification accuracy and computational efficiency. The GCFE method also requires fewer training parameters for deep-learning models than the traditional weighted Laplacian method, establishing its potential for more effective and efficient feature extraction in computer vision tasks. [ABSTRACT FROM AUTHOR]

Recent spectral graph sparsification research aims to construct ultra-sparse subgraphs for preserving the original graph spectral (structural) properties, such as the first few Laplacian eigenvalues and eigenvectors, which has led to the development of a variety of nearly-linear time numerical and graph algorithms. However, there is very limited progress for spectral sparsification of directed graphs. In this work, we prove the existence of nearly-linear-sized spectral sparsifiers for directed graphs under certain conditions. Furthermore, we introduce a practically-efficient spectral algorithm (diGRASS) for sparsifying real-world, large-scale directed graphs leveraging spectral matrix perturbation analysis. The proposed method has been evaluated using a variety of directed graphs obtained from real-world applications, showing promising results for solving directed graph Laplacians, spectral partitioning of directed graphs, and approximately computing (personalized) PageRank vectors. [ABSTRACT FROM AUTHOR]

Many important biological facts have been found as single-cell RNA sequencing (scRNA-seq) technology has advanced. With the use of this technology, it is now possible to investigate the connections among individual cells, genes, and illnesses. For the analysis of single-cell data, clustering is frequently used. Nevertheless, biological data usually contain a large amount of noise data, and traditional clustering methods are sensitive to noise. However, acquiring higher-order spatial information from the data alone is insufficient. As a result, getting trustworthy clustering findings is challenging. We propose the Cauchy hyper-graph Laplacian non-negative matrix factorization (CHLNMF) as a unique approach to address these issues. In CHLNMF, we replace the measurement based on Euclidean distance in the conventional non-negative matrix factorization (NMF), which can lessen the influence of noise, with the Cauchy loss function (CLF). The model also incorporates the hyper-graph constraint, which takes into account the high-order link among the samples. The CHLNMF model's best solution is then discovered using a half-quadratic optimization approach. Finally, using seven scRNA-seq datasets, we contrast the CHLNMF technique with the other nine top methods. The validity of our technique was established by analysis of the experimental outcomes. [ABSTRACT FROM AUTHOR]

An increasing number of measurement electrodes have been designed to satisfy the demand for high-resolution detection using galvanic logging technology in complex formations. The forward modeling response analysis of logging tools has important guiding significance in the design of galvanic logging tools. Based on a three-dimensional finite element numerical simulation method, we established a forward model of galvanic multi-electrodes in a complex formation. We also designed a symmetrical resistance network model of the formation with equivalent resistance between two electrodes. A symmetrical resistance network was derived using the balanced bridge method. The asymmetrical admittance matrix was extended to a symmetrical extended admittance matrix to realize a convenient calculation of the equivalent symmetrical resistance network in complex formations. Verification of the microcolumn-focused logging tool, with nine electrodes in a simulated standard well, and an evaluation of the degree of invasion in an actual oil well indicate that this calibration method can improve the measurement accuracy of galvanic logging instruments. [ABSTRACT FROM AUTHOR]

Partition problems in graphs are extremely important in applications, as shown in the Data Science and Machine Learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the second smallest eigenvalue a (G) of the Laplacian matrix L G of the graph G. This problem corresponds to the minimization of a quadratic form associated with L G , under certain constraints involving the ℓ 2 -norm. We introduce and investigate a similar problem, but using the ℓ 1 -norm to measure distances. This leads to a new parameter b (G) as the optimal value. We show that a well-known cut problem arises in this approach, namely the sparsest cut problem. We prove connectivity results and different bounds on this new parameter, relate to Fiedler theory and show explicit expressions for b (G) for trees. We also comment on an ℓ ∞ -norm version of the problem. [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices. For a given integer k such that 1 ≤ k ≤ n , the k -token graph F k (G) of G is defined as the graph whose vertices are the k -subsets of the vertex set of G , and two of them are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this article, we study the structural and spectral properties of token graphs. We describe the adjacency matrix and the Laplacian matrix of F k (G) and obtain bounds on the adjacency and Laplacian spectral radii of F k (G). Interestingly, it is found that S n , the star graph on n vertices, has the same Laplacian spectral radius as that of F k (S n). It was conjectured that for any graph G , the algebraic connectivity of F k (G) is equal to the algebraic connectivity of G. This result turned out to be a theorem, as it was proved by using the theory of the continuous Markov chain of random walks and the interchange process. However, proving this theorem using algebraic and combinatorial methods is still an open and interesting problem. Using combinatorial techniques, we prove that the theorem holds for a class of graphs that have a cut-vertex of degree n − 1. We also prove it by restricting the smallest degree of the k -token graph of G. [ABSTRACT FROM AUTHOR]

In this paper, a fully distributed strategy for the economic dispatch problem (EDP) in the smart grid is proposed. The economic dispatch model considers both traditional thermal generators and wind turbines (WTs), integrating generation costs, carbon trading expenses, and the expected costs associated with the unpredictability of wind power. The EDP is transformed into an equivalent optimization problem with only an equality constraint and thus can be solved by an alternating-direction method of multipliers (ADMM). Then, to tackle this problem in a distributed manner, the outer-layer framework of the proposed strategy adopts a parallel ADMM, where different variables can be calculated simultaneously. And the inner-layer framework adopts a finite-step consensus algorithm. Convergence to the optimal solution is achieved within a finite number of communication iterations, which depends on the scale of the communication network. In addition, leveraging local and neighbor information, a distributed algorithm is designed to compute the eigenvalues of the Laplacian matrix essential for the finite-step algorithm. Finally, several numerical examples are presented to verify the correctness and effectiveness of the proposed strategy. [ABSTRACT FROM AUTHOR]

In the field of brain tumor segmentation, accurately capturing the complexities of tumor sub-regions poses significant challenges. Traditional segmentation methods usually fail to accurately segment tumor subregions. This research introduces a novel solution employing Graph Neural Networks (GNNs), enriched with spectral and spatial insight. In the supervoxel creation phase, we explored methods like VCCS, SLIC, Watershed, Meanshift, and Felzenszwalb–Huttenlocher, evaluating their performance based on homogeneity, moment of inertia, and uniformity in shape and size. After creating supervoxels, we represented 3D MRI images as a graph structure. In this study, we combined Spatial and Spectral GNNs to capture both local and global information. Our Spectral GNN implementation employs the Laplacian matrix to efficiently map tumor tissue connectivity by capturing the graph's global structure. Consequently, this enhances the model's precision in classifying brain tumors into distinct types: necrosis, edema, and enhancing tumor. This model underwent extensive hyper-parameter tuning to ascertain the most effective configuration for optimal segmentation performance. Our Spectral–Spatial GNN model surpasses traditional segmentation methods in accuracy for both whole tumor and sub-regions, validated by metrics such as the dice coefficient and accuracy. For the necrotic core, the Spectral–Spatial GNN model showed a 10.6% improvement over the Spatial GNN and 8% over the Spectral GNN. Enhancing tumor gains were 9.5% and 6.4%, respectively. For edema, improvements were 12.8% over the Spatial GNN and 7.3% over the Spectral GNN, highlighting its segmentation accuracy for each tumor sub-region. This superiority underscores the model's potential in improving brain tumor segmentation accuracy, precision, and computational efficiency. [ABSTRACT FROM AUTHOR]

This paper established the idea of minimum evdominating matrix of semigraph and calculated its energy. The minimum ev-dominating energy EmeD(G) of a semigraph G is the sum of the absolute values of the eigenvalues of the minimum ev-dominating matrix. Here some results are also derived in connection with the energy of minimum evdominating matrix. Some lower bounds are also established. [ABSTRACT FROM AUTHOR]

Let G be a connected graph on n vertices with adjacency matrix A G. Associated to G is a polynomial d G (x 1 , ... , x n) of degree n in n variables, obtained as the determinant of the matrix M G (x 1 , ... , x n) , where M G = Diag (x 1 , ... , x n) − A G. We investigate in this article the set V d G (r) of non-negative values taken by this polynomial when x 1 , ... , x n ≥ r ≥ 1. We show that V d G (1) = Z ≥ 0. We show that for a large class of graphs one also has V d G (2) = Z ≥ 0. When V d G (2) ≠ Z ≥ 0 , we show that for many graphs V d G (2) is dense in Z ≥ 0. We give numerical evidence that in many cases, the complement of V d G (2) in Z ≥ 0 might in fact be finite. As a byproduct of our results, we show that every graph can be endowed with an arithmetical structure whose associated group is trivial. [ABSTRACT FROM AUTHOR]

The second largest eigenvalue of a graph is an important algebraic parameter which is related with the expansion, connectivity and randomness properties of a graph. Expanders are highly connected sparse graphs. In coding theory, Expander codes are Error Correcting codes made up of bipartite expander graphs. In this paper, first we prove the interlacing of the eigenvalues of the adjacency matrix of the bipartite graph with the eigenvalues of the bipartite quotient matrices of the corresponding graph matrices. Then we obtain bounds for the second largest and second smallest eigenvalues. Since the graph is bipartite, the results for Laplacian will also hold for Signless Laplacian matrix. We then introduce a new method called vertex-split of a bipartite graph to construct asymptotically good expander codes with expansion factor D/2 < α < D and ϵ < 1/2 and prove a condition for the vertex-split of a bipartite graph to be k-connected with respect to λ2. Further, we prove that the vertex-split of G is a bipartite expander. Finally, we construct an asymptotically good expander code whose factor graph is a graph obtained by the vertex-split of a bipartite graph. [ABSTRACT FROM AUTHOR]