An optimization approach to degree deviation and spectral radius

On the spectral radius of unbalanced signed bipartite graphs

Some Turán-type results for the signless Laplacian spectral radius

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 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.

Spectral radius and rainbow k-factors of graphs

A new comparison framework is developed to fairly evaluate and properly compare single-solve and sub-step time integration algorithms with controllable numerical dissipation. Unlike existing and/or past practices that rely only upon using different time step sizes to ensure comparable computational cost, this work emphasizes the necessity of simultaneously accounting for the effect of the infinity spectral radius, i.e., ρ ∞ ∈ [ 0 , 1 ] , which governs the amount of numerical dissipation in practical applications. Neglecting this consideration, as is common in much of the current literature, may result in less accurate conclusions and potentially overlook some aspects of the performance of traditional single-step single-solve time integration algorithms. Numerical implementations are presented to demonstrate how to compute spectral properties in the newly proposed comparison framework to achieve a fair and proper comparison. The numerical illustrations demonstrate that the two-sub-step ρ ∞ -Bathe method with ρ ∞ ∈ [ 0 , 0.1 ] exhibits improved numerical properties compared to the single-step single-solve KDP- α method. Likewise, the two-sub-step ρ ∞ -Bathe method with ρ ∞ ∈ [ 0 , 0.2 ] provides improvements compared to the single-step single-solve TPO/G- α method. However, outside these ranges, the ρ ∞ -Bathe method is inferior in the sense of spectral accuracy to the KDP- α and TPO/G- α methods. Additional comparisons involving three- and four-sub-step algorithms are also included to further highlight the trade-offs and advantages/disadvantages relative to single-step single-solve methods. Besides the theoretical formulations, the numerical analysis is additionally verified by the stiff spring–mass, hardening spring, and elastic spring-pendulum problems. Furthermore, other results from the literature are revisited for the proper comparison.

Training in reservoir computing (RC) requires using an initial part of the input sequence for the reservoir internal state to synchronize with the driver. The length of this transient, often treated as an empirical washout parameter, is a critical resource: too short and training fails, too long and data are wasted. These synchronization transients are quantified and explain when and why they become prohibitively long. For spectral radius ρ<1 (the largest absolute eigenvalue of the recurrent weight matrix), a closed-form upper bound is derived from the linearized dynamics that captures the dependence on tolerance and typical initial separation. Near and above unity, ensemble experiments reveal heavy-tailed transient-time distributions: most realizations synchronize quickly, yet a non-negligible fraction requires orders of magnitude longer washout. Non-linear saturation under driving shortens transients and can restore the Echo State Property even for ρ>1, with rates controlled by input amplitude and leakage. At zero input, persistence vs silence is dictated by topology: interpreting the recurrent weight matrix A as a directed graph and decomposing it into cycles explains which nodes sustain activity (core), which inherit it downstream (driven), and which die out (quiescent). This structure-dynamics correspondence yields practical design rules: pre-screen reservoirs by monitoring state distances; favor cycle-rich topologies or nodes downstream of cycles; avoid very sparse or very dense regimes at large ρ; and tune input scaling/leakage to exploit saturation. Overall, this turns "washout" from a heuristic into a measurable, engineerable property with clear levers for robust, data-efficient RC.

We extend the classical Perron–Frobenius theory to tensors that may have negative entries, thereby broadening the scope of spectral analysis beyond the nonnegative setting. Under certain sufficient conditions, we establish the existence of a Perron–Frobenius eigenpair for such tensors and characterize the corresponding spectral radius. As an application, we propose a novel concept termed Perron–Frobenius splitting for tensors, which facilitates the solution of multi-linear systems via tensor splitting iterative methods. This framework generalizes the regular and weak regular splittings of the tensors. Furthermore, we provide convergence analyses and comparison theorems for the Perron–Frobenius splittings of the tensors.

This paper investigates the fractal dynamical behavior of a discrete Caputo fractional-order hepatitis C virus model. First, we analyze the stability of the system by using spectral radius and design the fractional-order controller based on coordinate transformation. Then, a nonlinear coupling controller is constructed to achieve synchronization between two fractional-order models with different parameters and different fractional orders, and the synchronization is supported by rigorous mathematical proof. Numerical simulations are used to verify the effectiveness of control and synchronization.

Urban last-mile logistics systems must improve service responsiveness while reducing environmental impact. While micromobility-based delivery fleets offer significant emission advantages compared to conventional vans, their operational efficiency depends on adaptive, data-driven capacity allocation. We develop and analyze a real-time optimization framework that explicitly integrates sustainability considerations into zone-level fleet allocation decisions. The continuous-time backlog dynamics admit a closed-form discrete-time prediction, enabling computationally efficient rolling-horizon fleet reallocation. Sustainability is explicitly embedded through zone-specific emission factors and a multi-criteria objective function balancing backlog reduction, environmental impact, and operational stability. In a ten-zone numerical case study with a fleet of 40 vehicles, the proposed method reduced backlog in all zones within a 15-min interval while preserving strict feasibility and stability (spectral radius is less than 1). The framework also demonstrated a controllable emission–service trade-off via sensitivity analysis. These results suggest the practical applicability and real-time suitability of the proposed Industry 4.0-aligned optimization approach.

ABSTRACT Let and denote the edge connectivity and the maximum number of edge‐disjoint spanning trees contained in a graph , respectively. Fan, Gu, and Lin studied a tight spectral condition of a connected graph to guarantee . The famous theorem of Tutte and Nash‐Williams implies that and are closely related with , so it is interesting to explore conditions on a graph with to warrant . Since graphs with edge‐disjoint spanning trees are ‐edge‐connected, the converse does not hold. We initially provide a tight spectral radius condition to guarantee in ‐edge‐connected graphs with a unique extremal graph characterization. Moreover, an extensive spectral condition on in ‐edge‐connected graphs with fixed minimum degree is determined. For , we further study analogous results on 3‐edge‐connected graphs to ensure , which implies comprehensive spectral arguments on for .

Abstract We introduce a family of geometric anisotropic Banach spaces on Heisenberg nilmanifolds and study the spectrum of the composition operator associated to partially hyperbolic automorphisms. Choosing amongst the family of Banach spaces, it is possible to make the essential spectral radius arbitrarily small. We show that the exterior part of the discrete spectrum coincides with the spectrum restricted to the kernel of one of the operators associated to the nil-automorphism. Moreover we show that the remainder of the discrete spectrum is self-similar, it is given by scaled copies of the exterior part.

A graph is trivial if it contains one vertex and no edges. The essential connectivity κ ′ of G is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least two non-trivial components. Let A n κ ′ , δ be the set of graphs of order n with minimum degree δ and essential connectivity κ ′ . In this paper, we determine the graphs attaining the maximum spectral radii among all graphs in A n κ ′ , δ and characterize the corresponding extremal graphs. In addition, we also determine the digraphs that achieve the maximum spectral radii among all strongly connected digraphs with given essential connectivity and give the exact values of the spectral radii of these digraphs.

Abstract We establish some conditions under which $\textrm{GL}(d,\mathbb{R})$-valued cocycles over a subshift of finite type equipped with an equilibrium state exhibit exponential asymptotics for the spectral radius. Specifically, we show that the exponential growth rate of the spectral radius converges to the top Lyapunov exponent of the cocycle. This result provides a partial answer to a question posed by Aoun and Sert in [2]. Our approach relies on large deviation estimates for linear cocycles, which may be of independent interest.

Financial markets often display nonlinear and turbulent dynamics during periods of stress, and crude-oil and global equity systems frequently demonstrate closely connected forms of instability. Earlier studies report multifractality, chaotic features and regime-dependent spillovers across commodities and equities, yet existing approaches rarely succeed in capturing both the intrinsic complexity of oil-market behavior and the changing structure of cross-asset dependence. This limitation reduces the ability to distinguish calm from turbulent regimes and weakens short-horizon risk assessment. The present study introduces a unified framework that quantifies and predicts systemic instability within the coupled oil–equity system. The analysis constructs a crude-oil complexity index based on multifractal fluctuation analysis, permutation and approximate entropy, and Lyapunov-based indicators of chaotic dynamics. At the same time, it develops an information-theoretic network of global equity and energy-sector returns and summarizes its instability through measures of edge turnover, spectral radius, degree entropy and strength dispersion. These components are combined to form the Coupled Multiscale Chaos Index (CMCI), a scalar state variable that distinguishes calm, transitional and chaotic market regimes. Empirical results indicate that Brent and WTI exhibit pronounced multifractality, elevated entropy and positive Lyapunov exponents, while the dependence network becomes more centralized, more clustered and more capable of shock amplification during high-CMCI states. The CMCI moves closely with realized volatility and provides significant predictive content for five-day variance across major global equity benchmarks, with performance superior to models that rely only on macro-financial controls. Out-of-sample evaluation shows that forecasts incorporating measures of complexity record substantially lower MSE and QLIKE losses. The findings indicate that systemic instability reflects the interaction between local chaotic dynamics in crude-oil markets and turbulence in the global dependence network. The CMCI offers a practical early-warning indicator that supports risk management, forecasting and macroprudential supervision.

Abstract In this paper, we provide a new proof of a density version of Turán's theorem. We also rephrase both the theorem and the proof using entropy. With the entropic formulation, we show that some naturally defined entropic quantity is closely connected to other common quantities such as Lagrangian and spectral radius. In addition, we also determine the Turán density for a new family of hypergraphs, which we call tents. Our result can be seen as a new generalization of Mubayi's result on the extended cliques.