Spectral Radius Research Articles

Analysis of three dimensional advection-diffusion type models using wavelet scheme with error estimate

Abstract Solving high-dimensional partial differential equations (PDEs) is an essential problem in various fields of science and engineering. Analytical solutions for high-dimensional PDEs are often intricate and rare, as very few problems possess such solutions. Therefore, numerical methods play a pivotal role in overcoming these challenges. This work proposes an efficient numerical method for solving three-dimensional advection-diffusion equations. The present approach employs Haar wavelets to estimate the solution and its spatial derivatives, while the time derivative is approximated using a finite difference scheme. The collocation technique is then applied, transforming the three-dimensional PDEs into coupled linear algebraic equations with unknown wavelet coefficients. Solving this linear system provides the unknown coefficients, which refine the solution and its derivatives sequentially at each time step. Additionally, the error and stability analysis of the proposed numerical scheme are examined, linking the resolution level with the error and the spectral radius with stability. The method is implemented for the solution of three-dimensional advection-diffusion equations. To asses the attainments and performance of the method, L∞, L2, and LRMS, error norms along with the relative error associated with infinity and L2 norms are presented in tabulated form. Two- and three-dimensional solution profiles are also displayed to demonstrate the efficiency of the suggested technique. Simulations confirm that the proposed numerical method is a powerful tool for the numerical approximations of three-dimensional advection-diffusion type models.

The $$A_{\alpha }$$-spectral radius for $$\{P_{2}, C_{3}, P_{5}, \mathcal {T}(3)\}$$-factors in graphs

The $$A_{\alpha }$$-spectral radius for $$\{P_{2}, C_{3}, P_{5}, \mathcal {T}(3)\}$$-factors in graphs

Optimization Techniques in Machine Learning Models Using Banach Space Theory: Applications in Engineering and Management

Abstract: This paper explores the interplay between Banach space theory and machine learning optimization, offering novel theoretical insights with applications in engineering and management. We establish a suite of original theorems that bridge functional analysis and data-driven models, including: (1) convergence rates for gradient descent in reflexive Banach spaces under norm-attainability conditions, (2) operator norm bounds governing neural network generalization, and (3) adversarial robustness guarantees via Lipschitz continuity in non-Euclidean settings. Methodologically, we develop Banach-space analogues of fundamental results-from SVM duality to PID control stability-while demonstrating their utility in resource allocation and time-series forecasting through Orlicz space embeddings. Our framework not only extends classical optimization theory to infinite dimensional function spaces but also provides implementable regularization strategies for deep learning. The results are substantiated by rigorous proofs leveraging weak∗ compactness, spectral radius analysis, and semigroup theory. For practitioners, we derive explicit error bounds and convergence rates applicable to high-dimensional datasets and non-smooth objectives. This work thus unifies abstract functional-analytic concepts with modern machine learning challenges, offering new tools for both theoretical analysis and algorithmic design.

Domination number and (signless Laplacian) spectral radius of cactus graphs

A cactus graph is a connected graph whose block is either an edge or a cycle. A vertex set $S\subseteq V(G)$ is said to be a dominating set of a graph $G$ if every vertex in $V(G)\setminus S$ is adjacent to a vertex in $S$. There are several results on the (signless Laplacian) spectral radius and domination number in graph theory. In this paper, we determine the unique graph with the maximum adjacency spectral radius and signless Laplacian spectral radius among all cactus graphs with fixed domination number.

The distance Seidel matrix of connected graphs

For a connected graph G, we present the concept of a new graph matrix related to its distance and Seidel matrix, called distance Seidel matrix D S ( G ) . Suppose that the eigenvalues of D S ( G ) be ∂ 1 S ( G ) ≥ ⋯ ≥ ∂ n S ( G ) . In this article, we establish a relationship between the distance Seidel eigenvalues of a graph with its distance and adjacency eigenvalues. We characterize all the connected graphs with ∂ 1 S ( G ) = 3. Also, we determine different bounds for the distance Seidel spectral radius and distance Seidel energy. We analyze the effect of edge deletion on the distance Seidel energy of the complete bipartite graph. Moreover, we obtain the distance Seidel spectra of different graph operations such as join, cartesian product, lexicographic product, and unary operations like the double graph and extended double cover graph. We give various families of distance Seidel cospectral and distance Seidel integral graphs as an application.

Characterization of the BJ-Orthograph Radii in a Certain Class of C*-Algebra

In this paper, we investigate the properties of the radius of the BJ-orthograph within the context of finite-dimensional C ∗ -algebras. The BJ-orthograph is a combinatorial structure derived from the orthogonality relations between elements in a C ∗ -algebra, named after its conceptual roots in Birkhoff-James orthogonality. Our primary focus is to establish a comprehensive understanding of the radius of this orthograph, which serves as a measure of the ”distance” from the center to the furthest vertex in this graph-theoretic representation. We begin by defining the BJ-orthograph and outlining its construction in finite-dimensional C ∗ -algebras. Subsequently, we explore the mathematical framework necessary for analyzing its radius, including relevant graph-theoretic and algebraic concepts. Through a series of theorems and lemmas, we derive explicit formulas and bounds for the radius of the BJ-orthograph, leveraging properties unique to C∗ -algebras such as the spectral radius and norm properties. Our results demonstrate how the algebraic structure and dimensionality of the C ∗ -algebra influence the radius of the BJ-orthograph. We provide illustrative examples to highlight these relationships and discuss potential applications in quantum information theory and operator algebras. Finally, we propose several open questions and directions for future research, aiming to extend our findings to broader classes of C ∗ -algebras and other orthogonality-based graphs.

On the spectral radius and energy of the degree distance matrix of a connected graph

Abstract Let G G be a simple connected graph on n n vertices. The degree of a vertex v ∈ V ( G ) v\in V\left(G) , denoted by d v {d}_{v} , is the number of edges incident with v v and the distance between any two vertices u , v ∈ V ( G ) u,v\in V\left(G) , denoted by d u v {d}_{uv} , is defined as the length of the shortest path from u u to v v . The distance matrix of G G , denoted by, D ( G ) D\left(G) , is defined as D ( G ) = d u v D\left(G)={d}_{uv} . We now define and investigate the degree distance matrix of a connected graph G G , defined as M D D ( G ) = ( ( d u + d v ) d u v ) u , v ∈ V ( G ) {M}_{DD}\left(G)={\left(\left({d}_{u}+{d}_{v}){d}_{uv})}_{u,v\in V\left(G)} . In this article, first, we derive bounds for the largest eigenvalue of the degree distance matrix. Then, we establish some bounds for the energy of the degree distance matrix of G G .

Analysis of SLBPS node-based model by Schur complement technique

The paper proposes and analyses a node-based nonlinear mathematical SLBPS model that describes the virus and patch propagation. The model is developed by using continuous Markov chain equations. Two types of networks, virus spreading and patch dissemination, are used in a single network. The dynamics of the model are disclosed. The four equilibrium states exist: two are disease-free, and two are endemic. The linear stability of all equilibrium points is explored by using Schur's complement technique. The results show that the spectral radius of the patch-forwarding and the virus-spreading networks have a marked impact on the prevalence of computer viruses. Some key factors that influence the virus's prevalence are also revealed.

Learning Rate of the Model Algorithm for Iterative Learning Control in Lung Nodule Surgical Continuum Robot Systems

ABSTRACTDuring surgical operations, the distal end of lung nodule surgical robots is frequently confronted with diverse and intricate disturbances, thereby posing significant challenges for nonlinear control of such continuum robot systems. The continuum robot has a complex nonlinear dynamic model, and the coupling between the joints will affect each other, which makes the joint control of the continuum robot difficult. In addition, the motion of the continuum robot also needs a real‐time control strategy. Based on the above analysis, this paper proposes a nonlinear iterative learning method, which is grounded in model algorithmic learning rates, for the control of the distal end of a surgical robot utilized in pulmonary nodule operations. This method not only considers the control error and its higher derivative, but also includes the parameters of the system model. Then, based on the learning rate determined by the model algorithm and the actual control input from the current iteration, the control input for the next iteration is calculated, thereby advancing the iterative learning process. Finally, the stability of the entire nonlinear iterative learning process is proved by the spectral radius condition under the global Lipschitz condition. The effectiveness and robustness of the proposed method have been verified through MATLAB/Simulink, demonstrating high precision and superior performance.

Generalized distance spectral radius of some t-partitioned transmission regular graphs

Let [Formula: see text] be an [Formula: see text]-vertex connected graph with the distance matrix [Formula: see text] and the diagonal matrix of transmissions [Formula: see text]. The generalized distance matrix of a connected graph [Formula: see text] is [Formula: see text], where [Formula: see text], [Formula: see text]. Through this matrix one can study distance, distance Laplacian, distance signless Laplacian, and [Formula: see text]- spectra of [Formula: see text] in a unified way. A simple connected graph [Formula: see text] is called [Formula: see text]-partitioned transmission regular if the vertex set of [Formula: see text] can be partitioned as [Formula: see text] such that for any pair of indices [Formula: see text] (not necessarily distinct) in [Formula: see text] and [Formula: see text], [Formula: see text] is constant. In this paper, for the spider graph [Formula: see text] and the radar graph [Formula: see text], we find matrices of order [Formula: see text] (quotient matrices) corresponding to [Formula: see text] and [Formula: see text] such that the largest eigenvalues of these matrices are generalized distance spectral radius of [Formula: see text] and [Formula: see text], respectively. Moreover, we determine the full [Formula: see text]-spectrum of [Formula: see text] and [Formula: see text] (wheel graph).

B-spline wavelets series solution for the class of singular IVPs of fourth-order Emden–Fowler equations

In this research, the authors simulate a fourth-order Emden–Fowler model by developing the composite numerical algorithm with the composition of semi-orthogonal linear B-spline wavelets and Newton–Raphson (NR) method. In the present algorithm, spatial discretization is achieved through the utilization of compactly supported semi-orthogonal B-spline scaling and wavelet functions. Then the nonlinear system of equations is obtained and simulated by the NR method. The convergence of the NR method is demonstrated by the theorem of spectral radius. Further, error analysis of B-spline wavelets is discussed and prove that the error bound is negative exponentially proportional to the resolution level.

Spectral radii of arithmetical structures on cycle graphs

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer-valued vectors ( d , r ) such that ( diag ⁡ ( d ) − A G ) ⋅ r = 0 , where the entries of r have gcd 1 and A G is the adjacency matrix of G. In this article, we find the arithmetical structures that maximize and minimize the spectral radius of ( diag ⁡ ( d ) − A G ) among all arithmetical structures on the cycle graph C n .

SCGG: Smart City Network Topology Graph Generator

ABSTRACTSmart cities use information and communication technology to promote citizen welfare and economic growth within a sustainable environment. To guarantee that different urban actors, including people, devices, companies, and governments, can communicate efficiently, securely, and reliably, a robust, adaptable network infrastructure is required. However, the increasing complexity of the systems involved poses a challenge to smart city network modeling. Network topology generators produce synthetic networks that can reflect the underlying properties of real‐world networks, providing a practical approach to designing, testing, and implementing complex systems such as smart cities, yet the limited number of network topology generators for smart city applications has long prevented the proper development, investigation, and evaluation of various network configurations. In this article, a novel Smart City Network Topology Graph Generator (SCGG) is proposed to create a pseudorandom topology that mimics real smart city networks. The main goal of SCGG is to generate a network topology for smart cities that captures the interconnectivity of several communication technologies, such as wireless sensor networks (WSN), Internet of Things (IoT), and cellular networks. The SCGG system is characterized by the number of clusters, the average number of nodes, the number of layers, and the node density. The general network architecture and path‐related variables of the generated topologies are evaluated based on different graph theory measures, focusing on both global graph‐level characteristics and local node‐level features. The experimental results, demonstrating high natural connectivity and a low spectral radius value, offer a reliable tool for optimizing and strengthening the behavior and performance of smart city networks under different conditions to improve their robustness, minimize the probability of disruptions or failures, and enhance overall efficiency to ensure a resilient network.

Some Nordhaus–Gaddum-Type Results on Spectral Radius and Transversals of Uniform Hypergraphs

Some Nordhaus–Gaddum-Type Results on Spectral Radius and Transversals of Uniform Hypergraphs

An Input–Output Approach to Structured Stochastic Uncertainty in Continuous Time

ABSTRACTWe consider the continuous‐time setting of linear time‐invariant (LTI) systems in feedback with multiplicative stochastic uncertainties. The primary objective of this article is to establish conditions for mean‐square stability using a purely input–output approach, avoiding the need for state space realizations. This approach broadens the applicability to a wider range of models, including infinite‐dimensional systems and systems with delays. The input–output approach leads to uncovering new tools such as stochastic block diagrams, which are closely related to stochastic integral equations rather than the more commonly used stochastic differential equations. We explore various stochastic interpretations, including Itô and Stratonovich, and develop block diagram conversion schemes to transition between these interpretations. The MSS conditions are given in terms of the spectral radius of a matrix operator that takes different forms when different stochastic interpretations are considered.

Spatial and temporal dynamics of rabies with nonlocal dispersal in a heterogeneous environment: insights for control strategies in the Philippines

Rabies poses a significant threat to human health. To address the effects of heterogeneous behaviors on rabies transmission, we propose a nonlocal diffusive model in heterogeneous environment. By means of semigroup theory and uniform persistence theory, we analyze the well-posedness of solution and derive the basic reproduction number R0, identified as the spectral radius of the next generation operator P. Our findings indicate the disease-free equilibrium is globally asymptotically stable when R0<1, whereas rabies is uniformly persistent when R0>11$\\end{document}]]>. Numerically, the theoretical results are verified. Moreover, we integrate data to explore the influence of various parameters on rabies transmission in the Philippines. The results suggest that euthanasia of stray dogs and vaccination of owned dogs reduce the risk of rabies outbreaks and highlight that ignoring spatial heterogeneity and nonlocal dispersal may lead to an underestimation of rabies risk. The study provides a foundation for control strategies, emphasizing regional variations and dog activity patterns.

Fractional perfect matching and distance spectral radius in graphs

Fractional perfect matching and distance spectral radius in graphs

Enhanced Phase Optimization Using Spectral Radius Constraints and Weighted Eigenvalue Decomposition for Distributed Scatterer InSAR

Enhanced Phase Optimization Using Spectral Radius Constraints and Weighted Eigenvalue Decomposition for Distributed Scatterer InSAR

Non-invasive online detection of power circuit breaker timing and asynchronism via phase-space spectral radius analysis

Abstract Mechanical condition monitoring of power circuit breakers (CBs) is vital for predictive maintenance, enabling early detection of mechanical issues. However, traditional offline testing methods require the CB to be taken offline, which is inefficient and incurs high maintenance costs. This paper introduces a novel online monitoring scheme for detecting CB contact timing and asynchronism using an enhanced dual threshold method-phase space spectral radius (DTM-PSSR). Vibration signals are initially processed using the DTM for variational mode decomposition. The signal component with the highest correlation to the original signal is selected for PSSR reconstruction. The reconstructed energy spectrum identifies the energy peak, used to calculate asynchronism and contact timing. This method employs a single accelerometer to directly capture relevant parameters without the need to take the CB offline, providing a practical, non-invasive solution for online CB monitoring and streamlining the maintenance process.

Evaluating spectral cloud effective radius retrievals from the Enhanced MODIS Airborne Simulator (eMAS) during ORACLES

Evaluating spectral cloud effective radius retrievals from the Enhanced MODIS Airborne Simulator (eMAS) during ORACLES