Distance signless Laplacian spectral radius and toughness in graphs with given minimum degree and order

On the spectral radius of atom-bond maximum degree-based sum-connectivity matrix and its applications as molecular descriptors

Link false data injection attacks on multi-agent systems with attacked target nodes.

Spectral radius and perfect matching in graphs with given fractional property

Echo State Networks (ESNs) are a powerful tool for modeling complex nonlinear dynamics in data-driven control applications. However, their high dimensionality poses significant challenges for reliable deployment and further analysis. This work presents a comprehensive analysis of incremental Input-to-State Stability ( ) for leaky integrator ESNs with feedback connections, focusing on deriving practical, computationally affordable stability conditions. We demonstrate that a joint condition on standard spectral metrics provides a way to promote , serving as a viable alternative to finding Lyapunov functions through computationally expensive techniques during the hyperparameter optimization. Through extensive numerical studies, we characterize how hyperparameters influence both these spectral metrics and . Furthermore, we have extended the available formal guarantees to models reduced through Proper Orthogonal Decomposition (POD), in order to formally ensure stability in the finally deployed models. The practical utility of this approach is validated on a quadruple-tank benchmark system, where we successfully identify stable, reduced-order ESNs that maintain high prediction accuracy. Our results provide ESN practitioners a simple yet effective framework for both promoting and ensuring stability in their models, facilitating their application in control scenarios. • Incremental Input-to-State Stability ( ) in Echo State Networks (ESNs) is evaluated. • A method to promote based on spectral norm and spectral radius is proposed. • The influence of hyperparameters on is evaluated through grid-search experiments. • Reduction methods are used to make ESNs tractable through Lyapunov arguments. • A quadruple-tank benchmark is used to validate the proposed methods.

MT-product of tensors, spectral radius and walks of hypergraphs

Estimating the spectral radius of Bell-type operators via finite dimensional approximation of orthogonal projections

We propose a five-dimensional stochastic time-delay differential equation model that includes virus-to-cell infection, silent and active cell-cell transmissions and CTL immune response. Using mathematical methods, the non-degenerate five-dimensional stochastic delay differential equation model is transformed into a degenerate eight-dimensional stochastic differential equation model. The existence of a unique global positive solution is demonstrated. Then, by establishing a suitable Lyapunov function, we obtain the existence of a stationary Markov process when the stochastic CTL-activated reproduction number is greater than one. Additionally, we derive a critical condition for virus extinction using spectral radius analysis method and the law of large numbers theorem. Finally, through numerical simulations, we explore the impacts of random perturbations and cell-cell transmission on the dynamic behaviour of the model, and investigate the effect of time delays on T-cell count and viral load. Furthermore, ensemble simulations quantify viral clearance probabilities relative to the clinical detection limit, revealing that CTL-mediated immunity utilizes environmental stochasticity more efficiently than B-cell-mediated mechanisms to accelerate viral clearance.

Newman, Schneider and Shalev defined the entropy of a graded associative algebra A as H ( A ) = limsup n → ∞ a n n , where a n is the vector space dimension of the n’th homogeneous component of A. When A is the homogeneous quotient of a finitely generated free associative algebra, they showed that H ( A ) ≤ a 2 . Using some results of Friedland on the maximal spectral radius of ( 0 , 1 ) -matrices with a prescribed number of 1’s, we improve on this bound.

Recent theoretical studies on reservoir computing have shown that when the spectral radius of the adjacency matrix is sufficiently small, the dynamics of a reference system can be embedded in the reservoir space, enabling the reconstruction of dynamical invariants. However, reservoir models often reproduce time series accurately even when the spectral radius is relatively large, where the underlying mechanism is not well understood. In this study, we investigate the reconstruction of dynamical structures from the perspective of Lyapunov analysis using reservoir computing applied to the Hénon map. By comparing the Lyapunov spectrum of the reservoir dynamics with that of the reference system, we show that the reference dynamics are embedded in a low-dimensional inertial manifold in the reservoir space. We further demonstrate that the full Lyapunov spectrum of the reference system can be recovered by restricting the analysis to the tangent space of this manifold, even when the spectral radius is relatively large. These results clarify the geometric mechanism underlying the successful reconstruction of chaotic dynamics by reservoir computing beyond the regime where theoretical guarantees currently exist.

An optimization approach to degree deviation and spectral radius

Some Turán-type results for the signless Laplacian spectral radius

On the spectral radius of unbalanced signed bipartite graphs

This article develops a unified framework for the redefined Zagreb descriptors that combines graph theory, spectral analysis, and molecular structure-property applications. We study the basic redefined Zagreb descriptors together with their higher-order variants, introduce the associated weighted graph matrices, and investigate their spectral radii, energies, and related structural interpretations. For several standard graph families, including paths, cycles, complete graphs, stars, complete bipartite graphs, wheels, and friendship graphs, we derive explicit formulas and show how these descriptors reflect different degree patterns and connectivity structures. We also establish general bounds for the descriptors and their weighted spectral quantities, thereby clarifying their connections with classical degree-based indices and adjacency energy. To examine chemical relevance, we first consider a small set of anticancer drug-like molecules and use it as an analytical descriptor study. In this setting, the redefined Zagreb descriptors and their energy-based analogues are strongly associated with size-related physicochemical quantities, while the mixed higher-order descriptor [Formula: see text] shows the strongest relationship with the minimum universal force-field energy. This part of the study is intended to identify informative descriptor trends rather than to establish a fully validated predictive model. We then carry out a broader QSPR study on 100 alcohol compounds with 17 physicochemical endpoints under repeated leakage-safe grouped external validation. The results show that the redefined Zagreb descriptor family and its derivative forms provide strong predictive performance for many targets, especially those related to molecular size, volume, and critical-property behavior. The derivative descriptors are therefore chemically meaningful and useful, while the combined representation shows where complementary information can be gained. Overall, the redefined Zagreb framework emerges as a mathematically rich and chemically useful family of descriptors whose combinatorial, spectral, and predictive roles can be studied in a unified way.

This paper introduces a rigorous class of two-dimensional Markov-switching autoregressive moving-average (2D MS-ARMA) models for spatial lattice data exhibiting regime-dependent dynamics. The switching mechanism is governed by a latent causal Markov random field that drives spatial transitions between regime-specific autoregressive and moving-average structures. We provide sufficient conditions for the existence of a strictly stationary solution through the top Lyapunov exponent associated with a sequence of random matrices obtained from a state-space representation constructed along the lexicographic order. For the first-order bidirectional specification, we derive explicit spectral conditions linking stationarity to the regime-dependent spectral radii. Sufficient conditions ensuring the existence of finite second-order moments are also provided. Parameter estimation is carried out using a variational expectation–maximization (VEM) algorithm based on a mean-field approximation of the posterior distribution of the hidden regimes. The E-step yields closed-form coordinate ascent updates, while the M-step relies on gradient-based numerical optimization with derivatives computed via recursive differentiation. Under increasing-domain asymptotics, we discuss the consistency and asymptotic behavior of the variational estimator. The proposed framework fills a methodological gap between classical one-dimensional Markov-switching ARMA models and spatial autoregressive structures by extending regime-switching theory to multi-indexed processes with rigorous probabilistic foundations. It provides a comprehensive basis for statistical inference, model diagnostics, and prediction in spatially heterogeneous environments.

Characterizations of fractional factor-critical graphs via size and spectral radius

Let Nn (with n ≥ 2) be the family of all nonnegative n × n matrices A = [aij ], where a11 = 0 and the remaining entries aij ∈ [0, 1) with a spectral radius ρ(A) = 1. We can establish a lower bound for the additional spread s(A) ≥ k/ n−1 , where k is the count of zero diagonal elements in matrix A. Furthermore, if matrix A possesses only two distinct eigenvalues, then it follows that s(A) ≥ n−2/n−1 . Additionally, we derived a few other lower bounds under a special family of matrices.

Abstract We consider a subshift of finite type endowed with a Markov measure that is given by a stochastic matrix. We introduce a Markov hole determined by a finite collection of allowed words in the subshift. We first present a simple yet precise formula to compute the escape rate into the hole as the spectral radius of a perturbed stochastic matrix, where the rule of perturbation is governed by the hole. The combinatorial nature of the subshift comes to our aid in obtaining another formulation of the escape rate as the logarithm of the smallest real pole of a certain rational function, by way of recurrence relations. This proves crucial in comparing the escape rates into cylinders based at words of fixed length. Merits of both the formulas are illustrated through examples.

ABSTRACT This paper analyzes the endemic spreading of COVID‐19 among co‐circulating respiratory infections by constructing a compartmental mathematical model of human and environment interactions. The mathematical structure describes the coupled interaction between the human population and the polluted viral environmental compartment. The existence, uniqueness, positivity, and boundedness of solutions are analyzed to confirm the well‐posedness of the mathematical formulation from the viewpoint of mathematical analysis and biological relevance. This paper embraces locally asymptotically stable disease‐free steady state under the condition of and unstable under the condition of . On the other hand, conditions of disease persistence steady state for both local as well as global asymptotical stability are incorporated under the conditions that spectral radius of next‐generation number yields greater than unity. Specifically, the local sensitivity computational analysis is implemented targeting the core parameters determining the communicating ability of the disease. Finally, the optimal control approach is introduced through the intervention of preventive actions, treatment of sick people, and environment decontamination. Computational results are implemented using MATLAB and cost‐effectiveness analysis of different action plans are carried out to explore the effectiveness of concerted action plans combining prevention, treatment, and environmental cleaning.

Transmission of tuberculosis (TB) among human population depends on an individual's infectiousness, which is further determined by the concentration of Mycobacterium tuberculosis (Mtb) in the body. Additionally, Mtb is resistant to dryness, cold, acidic, and alkaline environments and can survive in acidic and alkaline environments for 4-5 years. Mtb in the environment plays a significant role in TB transmission and should not be overlooked. To investigate the epidemiologic relationships among pathogens, hosts, and the environment, we first develop a multiscale TB model that includes multiple transmission routes (human-to-human and environment-to-human) and links Mtb-immune response interactions to TB transmission in population. We comprehensively analyze the dynamic properties of the fast system, slow system, and full system. Analysis results reveal that coupling bacterial processes within-host with transmission mechanisms between-host can trigger diverse complex behaviors, including both forward and backward bifurcation phenomena. This implies that thresholds routinely used to control TB infection or eliminate Mtb from an epidemiological or immunological perspective may fail under specific conditions; that is, even if the basic reproduction number is less than 1, endemic equilibria may still exist in the system. Second, from a microtherapeutic point of view, we establish an impulsive time-delayed differential equation to characterize the actual medication regimen for TB. The basic reproduction number is defined as the spectral radius of a linear integral operator. Then, we show that is a critical parameter that determines the persistence of the model. More precisely, if , the disease-free periodic solution is globally attractive; if , the disease is uniformly persistent. Finally, we employ numerical methods to elucidate the interactions between population transmission dynamics and pathogen dynamics. Specifically, the basic reproduction number of the full system increases rapidly with the rise in Mtb release rate, while its change is relatively slower with an increase in the immune rate. These results highlight the dominant role of chemotherapy, with immunotherapy playing only a supporting role.