The allocation of collective benefits and costs represents a central challenge for individuals and organizations operating in uncertain environments, including interval and fuzzy data scenarios. Cooperative games under uncertainty provide a versatile game-theoretical framework and robust analytical tools to address these challenges, offering insights into decision-making and resource distribution. This survey reviews the current state of research in this emerging field, highlighting recent theoretical developments, methodological advances, and computational approaches. Furthermore, it examines how cooperative games under uncertainty extend classical cooperative game theory and discusses their practical applications and potential across economic decision-making, operations research, and strategic planning contexts. By synthesizing existing literature, this work aims to provide a comprehensive understanding of both the theoretical foundations and applied implications of these models, offering guidance for future research directions.

Extracting multi-level information in colony images facilitates analysis and identification tasks of biomedical informatics. In order to achieve multi-level segmentation in colony images with multiple contrast levels, a closed-loop neural network model based on the stochastic resonance (SR) mechanism of neurons is proposed. First, this paper realizes the detection of transition pulses in sinusoidal, non-periodic bipolar binary signals, and one-dimensional strong and weak transition signals with multiple amplitude values. Then, through enhancement processing for the detection by the SR-based closed-loop neural network model, combined with the coati optimization algorithm, multi-target detection is achieved. Eventually, it can be applied to the segmentation of multi-level grayscale signals in two-dimensional images. Experimental results show that the proposed method can simultaneously detect strong and weak contrast edges, enrich image details, highlight image contours, and enhance the hierarchical sense of image edges, while exhibiting strong robustness against external noise. As a result, the proposed method provides a novel research framework for multi-contrast grayscale image segmentation under strong noise background.

The Burgers’ equation is a key model equation in fluid mechanics and nonlinear physics, widely used to describe the formation and propagation of shock waves. This paper proposes a highly accurate numerical algorithm to solve the two-dimensional Burgers’ equation. In the algorithmic framework, the time derivative is discretized using a fourth-order backward finite difference approach. In contrast, the spatial derivatives are processed by a fourth-order Padé scheme and a fourth-order compact finite difference scheme, respectively. To linearize the nonlinear term in the equation, an extrapolation-based linearization technique is introduced, which effectively reduces the computational complexity while maintaining the numerical accuracy. Theoretical analyses of this algorithm demonstrated its unique solvability and stability. Numerous numerical experiments verify the scheme’s stability, computational efficiency, and convergence. This approach achieves a fourth-order accuracy in both the temporal and spatial dimensions, thus offering the benefit of maintaining high agreement between the numerical and exact solutions, even when using relatively large time steps, thereby significantly improving the computational efficiency. In addition, the algorithm employs linearization to ease the computational demands, thereby providing a viable and efficient scheme to address high-dimensional nonlinear Burgers’ equations.

This study concentrates on the Allen–Cahn equation with logarithmic free energy, thereby proposing a second–order accurate finite difference scheme. The core design of the scheme adopts a linearized modified version of the classical leapfrog scheme for temporal discretization, combined with central differencing for spatial discretization. To further improve the numerical stability, a second–order stabilization term is systematically integrated into the discrete framework. A theoretical analysis reveals that under appropriate constraints on the time step size and stabilization parameter, the numerical solution strictly adheres to the maximum bound principle. A comprehensive stability analysis is performed in the maximum norm, and the corresponding error estimates are rigorously derived. Finally, numerical experiments are conducted to verify the correctness and effectiveness of the theoretical results.

We intend to study a class of the critical Schrödinger–Bopp–Podolsky system with p-Laplacian in R³. Under different perturbation terms, the existence and multiplicity of nontrivial solutions are obtained by using critical point theorem. Considering the influence of the p-Laplacian operator and critical and nonlocal terms, which cause the loss of the compactness condition, we attempt to address this difficulty by using the concentration-compactness principle and some clever analysis.

We consider a class of p-Laplace quasilinear Schrödinger Equations

where $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with regular boundary, 1 < p < ∞, 0 < γ < 1, 2 p−1 < q ≤ 2·p^*−1 for p ≤ N, 2p − 1 < q < ∞ for p > N, where $p^{*}=\frac{N p}{N-p}$ if 1 < p < N,p^* ∈ (p, ∞) is arbitrarily large if p =N, and p^* = ∞ if p > N. We establish global existence and multiplicity of positive solutions via a new strong comparison principle and a regularity result for weak solutions.

Contactless monitoring of respiration and heart rate in shared indoor spaces-such as hospital wards, eldercare facilities, and sleep labs-demands solutions that are unobtrusive, privacy-preserving, and robust to static reflections, multipath, and inter-subject interference. This paper presents a complete millimetre-wave frequency-modulated continuous-wave (FMCW) technology - multiple-input multiple- output (MIMO) radar framework for multi-subject vital-sign sensing with three key contributions: (i) clutter-robust target discovery to stabilize detections in realistic indoor scenes; (ii) high-resolution spatial separation to suppress inter-subject leakage; and (iii) hybrid time–frequency decomposition to improve heartbeat isolation by mitigating respiration harmonics and noise. Experiments on single-, two-, and three-subject datasets achieve Mean Absolute Error/ Root Mean Squared Error (MAE/RMSE) of 2.81/4.13 Beats Per Minute (BPM) for respiration and 2.43/3.61 BPM for heart rate. Compared with filtering-, Ensemble Empirical Mode Decomposition (EEMD), Compressive Sensing - Orthogonal Matching Pursuit (CS-OMP), and mmRH-Millimetre-wave radar heart-rate (mmRH) based baselines, the proposed framework yields substantially lower heart-rate errors while maintaining reliable multi-subject operation in shared indoor environments.

The objective of this paper is to investigate the local well-posedness of analytic solutions to the three-dimensional (3D) magnetohydrodynamic (MHD) boundary layer equations without structural assumptions. Specifically, the general initial data are required to be real-analytic in the tangential variables (x,y) and satisfy Sobolev regularity in the normal variable z. We first adopt a variable transformation involving ϕ(z) to homogenize the boundary conditions and eliminate resulting high- order terms. Additionally, we employ delicate energy estimates combined with Gauss weight functions to control linearly growth terms.

This study presents a hybrid evolution strategy–soft actor-critic (ES–SAC) controller for a reduced-degrees of freedom spatial (3D) humanoid gait model with predominantly sagittal-plane motion that focuses on arm-leg coordination. Policies are trained on flat terrain only and then evaluated on both flat and uneven ground under identical simulation settings. Arm-leg coordination is examined systematically in three modes (counter-phase with the legs [normal], in-phase [anti-normal], and fixed [passive]), and the results are compared with findings from human experiments. Whereas most prior studies evaluate policies primarily via reward curves, this work conducts a deep analysis using interpretable metrics aligned with human-like walking: speed-normalized power and torque, lateral and vertical deviations, and moment-balance terms. Simulation outcomes are reported quantitatively through these metrics rather than reward alone. Across five random seeds, a clear terrain-dependent trade-off appears between the swinging strategies: anti-normal attains higher forward speed and lower torque-per-speed, whereas normal provides better lateral tracking and lower power-per-speed on rough ground. Directional trends agree with human experiments (e.g., immobilized or in-phase arms raise metabolic cost), while numerical gaps reflect that the simulator measures mechanical power rather than metabolic energy. Within this framework, the impact of coordinated arm swing on balance and efficiency is quantified with a breadth and clarity uncommon in the literature.

This paper focuses on achieving prespecified-time stability in maglev suspension systems. Within a sliding mode control framework, we design three types of controllers for finite-time, fixed-time, and prespecified-time convergence. Pole assignment techniques are utilized to precisely tailor the system dynamics on the sliding manifold. The theoretical findings are validated through numerical simulations, which highlight the practical benefits of the proposed control strategies.

Autonomous navigation in complex, dynamic, and unstructured environments poses a major challenge for mobile robots due to limitations in traditional methods. Inspired by rodent entorhinal-hippocampal-prefrontal circuits, we proposed a brain-inspired navigation framework that leverages specialized spatial cells. The framework integrates entorhinal-hippocampal path integration from self-motion cues with prefrontal action selection via Sn-Plast modulated spiking neural networks (SNNs) to form initial goal-directed habits. Upon reaching the goal, CA1 place cells self-organize to optimize navigation routes, generating supervisory firing rate sequences that consolidate refined routes in the SNNs via feedback, enabling stable habits and rapid reshaping under task changes. Additionally, a chaining strategy decomposes complex long-distance tasks into sequential subtasks, enabling independent local habit formation and seamless integration into globally efficient routes. 3D robot simulation experiments showed superior performance over baseline algorithms in convergence speed and route efficiency. Moreover, the method exhibits rapid habit reshaping under dynamic task changes, reducing exploration episodes by 52.3%–62.2% and shortening route length by 5.6%–7.7% in multi-compartment environments via a hierarchical chaining strategy. Moreover, the framework can operate in real time at approximately 12 Hz, demonstrating its feasibility for robotic applications.

In this paper, we consider a trilinear weighted Hardy-Littlewood-Sobolev inequality as follows:

which can be regarded as the natural trilinear form of the Stein-Weiss inequality. For 1 ⩽ p₁, p₂, p₃ ⩽ ∞, we systematically establish the trilinear Stein-Weiss inequality. In particular, when 1 < p₁, p₂, p₃ < ∞, by means of appropriate space decomposition, we give the necessary and sufficient conditions to characterize the trilinear Stein-Weiss inequality.

In the present manuscript, we investigate the geometric behavior of complete and stochastically complete Cotton solitons isometrically immersed in a generalized Robertson-Walker (GRW) spacetime whose sectional curvature of the Riemannian fiber satisfies suitable curvature constraints. In this configuration, by assuming appropriate hypotheses on the warped structure of the GRW spacetime, we jointly use a Bochner-type formula with several maximum principles, integrability conditions, and a parabolicity criterion in order to show that such a Cotton soliton must be trivial and locally conformally flat. Additionally, applications of our main results for the steady state space, the de Sitter space, the future temporal cone of the Lorentz-Minkowski space, and a specific open subset of the anti-de Sitter space are also presented.

Graph contrastive learning methods have recently emerged as a promising solution to tackle the problem of label scarcity in real-world scenarios. However, most of the existing methods are still flawed due to the lack of guided objectives. Moreover, they fail to effectively utilize complementary structural information from different graphs. To address these limitations, we propose a novel semi-supervised teaching graph contrastive network (STGCN) for node classification. Based on the teaching network architecture, STGCN establishes multi-level contrastive objectives, ensuring rich and detailed supervision for graph encoders. Specifically, after carefully analyzing the intrinsic correlation between different augmented views, we send a diffusion graph and two augmented views together into the novel teaching network, which owns one teacher encoder to guide two shared student encoders. Furthermore, we introduce a random sampling mixing module that extracts complementary information from multiple graphs, along with a label propagation technique to fully exploit limited labeled data. Finally, our method incorporates supervised contrastive loss and node similarity regularization to ensure coherent alignment between labeled and unlabeled nodes. Extensive experiments on five real-world node classification datasets demonstrate a maximum of 2.60% higher improvement than other models.

A new mathematical model based on differential equations and deep learning is proposed to study the propagation mechanism of southern corn rust (SCR). Equilibrium and stability analyses of system are conducted in detail. Compared with the quantitative assessment of SCR, numerical parameter estimation and the corresponding interpretation are given.

In this paper, a data-driven robust distributed control strategy is proposed for cooperative voltage regulation of the direct-current (DC) microgrids such that the output voltages of all distributed generation units (DGUs) track the specified reference signal. Unlike in decentralized control where all DGUs need to know the information of the reference signal, each DGU only needs to obtain the information of its neighbors in the proposed distributed control strategy. In the presence of the unknown parameters of DGUs, a data-driven robust control method is proposed to achieve cooperative voltage regulation under the available data affected by noise. It is shown that the cooperative voltage regulation problem can be transformed into a local stabilization problem based on the proposed distributed control protocol. To address the local stabilization problem, the noisy data from local DGU and the constructed auxiliary system is collected, and then we develop a data-driven control framework that guarantees robust closed-loop stability under bounded noise. We formulate a data-dependent linear matrix inequality (LMI) based on D_Q-stabilization theory. This LMI enforces all eigenvalues of the closed-loop system to lie within a specified region inside the unit disk, thereby quantifying robustness through guaranteed stability margins when the collected data is corrupted by noise. The resulting gain matrices of the distributed control protocol are computed directly from noisy-data-dependent LMI without requiring any explicit model knowledge. Several simulation results are provided to show the effectiveness and robustness of the proposed methods.

This work investigates the dynamics of a diffusive viral infection model that incorporates a nonlinear incidence and follicular dendritic cells (FDCs). First, the well-posedness is established. Then, We show that the basic reproduction number serves as a critical threshold for global stabilities: the infection-free steady state is globally asymptotically stable when the basic reproduction number is less than one, while the model exhibits a uniform persistence when the basic reproduction number is greater than one. Under the condition that the basic reproduction number equals to one, the infection- free steady state is shown to be globally asymptotically stable given certain additional assumptions. Furthermore, the global stability of the infected steady state is established for the homogeneous case. We find that ignoring the spatial heterogeneity in the infection capacity of viruses and infected cells may lead to an underestimation of the transmission risk. Although the spatial heterogeneity of a FDC does not affect the basic reproduction number, neglecting the infection originating from a FDC may lead to an underestimation of the infection risk.

In the category of topological Abelian groups, we consider the usual notion of extension E = (B → X → A) of B by A and the notion of weak-split extension (when X → A has a continuous split A → X). Given a weak-split extension E, the topological Abelian group X is homeomorphic to B ×A, but in general,X need not be algebraic isomorphic to B ×A. In this paper, for two topological Abelian groups A,B, we study the Abelian group $E_{\mathbf{T A}}^{w s}$(A,B) of weak-split extensions of B by A modulo extension isomorphisms. We prove that $E_{\mathbf{T A}}^{w s}$(A,B) can be described as the Abelian group of all possible continuous sums given in the product topological space B×A (modulo topological isomorphism) having B as a topological subgroup and A as a topological quotient. We also give an alternative description of $E_{\mathbf{T A}}^{w s}$(A,B) as a quotient $\mathcal{Z}_{c}(A, B) / \mathcal{B}_{c}(A, B)$, where $\mathcal{Z}_{c}(A, B)$ are cocycles represented by certain continuous maps of the form A ×A → B, and similarly for the coboundaries B_c(A,B). For two topological Abelian groups A,B, we compare the Abelian group of ws-extensions $E_{\mathbf{T A}}^{w s}$(A,B) with the Abelian group of standard extensions E_A(A,B) where now A,B also denote the subjacent Abelian groups. We relate these different types of extensions using an exact sequence with six terms. Although the Bohr topology of discrete Abelian groups has been investigated by many workers, there still remain many parts that are not well understood. Here, as an application of the methods developed in the paper, new examples of nontrivial ws-extensions for discrete Abelian groups equipped with the Bohr topology are provided and some related open questions are also proposed.

Imbalanced node classification is an important research topic in graph learning. With the rise of deep learning, powerful models like graph neural networks (GNNs) have been widely used in node classification tasks and achieved promising performance. However, real-world graphs are usually imbalanced, characterized by some classes having an adequate amount of data while others lack data. This imbalance presents the suboptimal classification performance of the model. Therefore, research on the GNNs-based imbalanced node classification is crucial. This article aims to systematically summarize the development status of GNNs in imbalanced node classification. First, the related concepts of GNNs and imbalanced node classification are introduced to establish a solid foundation for readers. Second, the methods are divided into data-level methods and algorithm-level methods, then subdivided into seven subcategories. Especially, we discuss the key thoughts, relative strengths, and weaknesses of classic methods in each subcategory. Then, datasets and common evaluation metrics are compiled to provide a wealth of useful resources. Finally, future research directions for imbalanced node classification on graphs are introduced to promote the boom of this field.

This study investigates boundary value problems for nonlinear fractional differential equations. The differential operator is interpreted in the Hadamard sense and is coupled with a nonlinear term that involves the fractional derivative of the unknown function. The existence and multiplicity of positive solutions were established by a reducing-order technique based on the Guo–Krasnoselskii fixed-point theorem. Also, some examples are presented to illustrate the validity of our main results.

This study investigates a modified Bouali model (referred to as the business cycle model) through the lens of transition dynamics theory. Under reasonable theoretical assumptions, we identify the output-capital ratio c as the key parameter that determines the transition and chaos in the business cycle model. Through rigorous mathematical analysis, we demonstrate that this model exhibits both jump (discontinuous) and continuous transitions as the output-capital ratio c increases and crosses a critical value. In particular, for the output-capital ratio large enough c, the business cycle model develops a strange attractor, which means that there exists chaos in this business cycle model.

Systolic murmurs are extra heart sounds that occur during the contraction phase of the cardiac cycle, often indicating heart abnormalities caused by turbulent blood flow. Their intensity, pitch, and quality vary, requiring precise identification for the accurate diagnosis of cardiac disorders. This study presents an automatic classification system for systolic murmurs using a feature extraction module, followed by a classification model. The feature extraction module employs complex orthogonal matching pursuit to project single or multiple murmur segments onto a redundant dictionary composed of multiresolution complex Gabor basis functions (GBFs). The resulting projection weights are split and reshaped into variable-resolution time–frequency feature matrices. Processing multiple segments of a single recording using a shared dictionary mitigates murmur variability. This is achieved by learning the weights for each segment while enforcing that they correspond to the same set of basis functions in the dictionary, promoting consistent time–frequency feature matrices. The classification model is built based on a vision transformer to process multiple input matrices of different resolutions by passing each through a convolutional neural network for patch tokenization. All embedding tokens are then concatenated to form a matrix and forwarded to an encoder layer that includes multihead attention, residual connections, and a convolutional network with a kernel size of one. This integration of multiresolution feature extraction with transformer-based feature classification enhances the accuracy and reliability of heart murmur identification. An experimental analysis of four types of systolic murmurs from the CirCor DigiScope dataset demonstrates the effectiveness of the system, achieving a classification accuracy of 95.96%.

This paper investigates the adaptive tracking control problem for a class of strict-feedback uncertain nonlinear multi-agent systems (MASs) with input dead zones under the simultaneous presence of deception attacks and denial-of-service (DoS) attacks. A distributed leader-state estimator based on sampled data with time delays is proposed to estimate the leader’s state under DoS attacks. To overcome the nonexistence of higher-order derivatives of the leader’s estimated state, a filter is proposed and implemented. Based on the attacked output signals, a fuzzy state observer is constructed to reconstruct the unmeasurable states of the system. A fuzzy adaptive dead-zone control scheme is designed based on the backstepping method to mitigate the adverse effects of DoS attacks, deception attacks, dead zones, and uncertain nonlinear dynamics, while enabling the system to achieve the tracking control objective. Through Lyapunov stability analysis, the proposed control scheme is proven to guarantee that all signals in the closed-loop system are bounded and the tracking error converges to a neighborhood around the origin. Finally, simulations are conducted to verify the effectiveness of the theoretical results.

We study conformal minimal immersions into Rⁿ via the classical correspondence with holomorphic null curves in Cⁿ. After recalling a convenient Weierstrass-type parametrization of null data on simply connected domains, we present an explicit integral-free construction that produces minimal immersions directly from a single holomorphic seed function, meaning that the coordinate functions are obtained through closed-form algebraic expressions involving derivatives of the seed rather than through path integration of null data. This viewpoint leads to a reconstruction identity for the seed and gives concrete formulas for the induced metric and the associated Gauss map. For polynomial seeds, we obtain explicit families in arbitrary codimension with closed-form conformal factors. As an analytic application, we derive a second-variation formula for the area under holomorphic perturbations of the seed, expressed in terms of the third derivative of the perturbation. The discussion is local and formula-driven: we do not consider global period problems, completeness, embeddedness, or topological classification, as our goal is to develop explicit analytic constructions rather than a global classification theory.

This paper studies the existence of mild solutions for impulsive stochastic fractional integro-differential equations with finite delay and weakly singular kernels in separable Banach spaces. The model involves a Caputo derivative of order α ∈ (¹₂, 1), a cylindrical Wiener process, instantaneous impulses, and a singular kernel (t − s)^−β with β ∈ (0, 1 − α). To the best of our knowledge, the combined presence of impulsive effects, stochastic noise, finite delay, and weakly singular kernels has not yet been analyzed in the literature within a Caputo fractional framework in Banach spaces of type 2. Using resolvent families, Itô calculus in Banach spaces, and Krasnoselskii’s fixed point theorem, we establish the existence of mean-square mild solutions under natural growth and continuity assumptions. An example illustrates the applicability of the results.

By using the delicate energy method, we prove the global existence and stability of solutions to an inviscid compressible Oldroyd-B model near an equilibrium state in a periodic domain. Especially, the result justify the stabilization effect of elasticity on the compressible flows.

In this paper, we propose normalized time-fractional diffusion equations in two and three spatial dimensions, where the normalization guarantees that the total memory weight remains equal to one and allows a consistent interpretation of memory effects for different fractional orders. An efficient Fourier spectral method in space combined with a finite difference approximation in time is used to solve the governing equations in both two- and three-dimensional (2D and 3D) settings. A rigorous error analysis shows that the proposed scheme achieves a temporal convergence rate of order O(∆t^2−α), and numerical experiments confirm exponential accuracy in space. Extensive 2D and 3D numerical tests demonstrate the robustness of the method, and the numerical results show that smaller fractional orders slow down the evolution dynamics due to stronger memory effects. A comparison with the standard Caputo time-fractional diffusion model indicates that the normalized formulation provides a more uniform temporal decay behavior while preserving the essential dynamics. These computational results suggest that the proposed method offers an accurate and reliable numerical algorithm for simulating multidimensional diffusion processes with memory effects.

Symbolic functions, as an important class of nonsmooth functions, play a key role in many fields such as system control and optimization theory. In this study, the special case of nonlinear optimization problems with sign function constraints is discussed in depth. Based on the smooth approximation theory, the study first adopts an approximate substitution method to smooth the nonsmooth constraints and then transforms them into differentiable optimization problems, thus effectively avoiding the numerical computational difficulties caused by the sign function. Second, the constrained optimization problem is transformed into an unconstrained optimization problem by constructing an exact penalty function. Third, two forms of second-order dynamical systems are established to solve the problem, and a rigorous theoretical analysis of the stability of these systems is conducted. Finally, the convergence and computational efficiency of the proposed method are verified by numerical simulation experiments of the system, and the simulation results fully prove the effectiveness and practicality of the algorithm.

The aim of this paper is to investigate the delay-dependent stability of highly nonlinear hybrid neutral stochastic differential delay equations (NSDDEs). Departing from most existing studies, the system under consideration incorporates a time-varying delay that is not required to be differentiable. A novel decomposition scheme for the drift coefficient is introduced, relaxing the conventional restrictive Lipschitz condition on the delay component. By constructing appropriate Lyapunov functionals and employing an M-matrix approach, delay-dependent conditions are derived to ensure moment stability for the considered highly nonlinear NSDDEs. Finally, an example is given to demonstrate the effectiveness of our new theory.

Operator learning has become a powerful tool for accelerating the solution of parameterized partial differential equations (PDEs), enabling rapid prediction of full spatiotemporal fields for new initial conditions or forcing functions. Existing architectures such as the deep operator network (DeepONet) and the Fourier neural operator (FNO) show strong empirical performance, but often require large training datasets, lack explicit physical structure, and may suffer from instability in their trunk-network features, where mode imbalance or collapse can hinder accurate operator approximation. Motivated by the stability and locality of classical partition-of-unity (PoU) methods, we investigate PoU-based regularization techniques for operator learning and develop a revised formulation of the existing POU–PI–DeepONet framework. The resulting physics-informed partition penalty deep operator network (PIP² Net) introduces a simplified and more principled partition penalty that improves the coordinated trunk outputs, which leads to more expressiveness without sacrificing the flexibility of DeepONet. We evaluate PIP² Net on three nonlinear PDEs: the viscous Burgers equation, the Allen–Cahn equation, and a diffusion–reaction system. The results show that it consistently outperforms DeepONet, PI-DeepONet, and POU-DeepONet in prediction accuracy and robustness.

We have introduced the notion of quasi-resolving subcategories in a triangulated category with a proper class of triangles. We provided some criteria for computing the resolution dimension of objects relative to a quasi-resolving subcategory. We also provided a method to construct quasi-resolving subcategories from given ones.

In this paper, we consider the L1/ finite element methods for solving time-fractional Keller- Segel equations with a singular solution. For time direction, the L1 scheme under the nonuniform mesh is considered to approximate the Caputo derivative to handle the singularity of the solution. To further reduce computational storage requirements, we consider a fast L1 scheme based on the sum-of- exponentials skill for calculating fractional derivatives. Then, we derive a fully implicit discrete scheme by combining the finite element method in the spatial direction. Subsequently, unconditional stability and α-robust error estimates of the fully discrete scheme are derived. Finally, numerical examples are presented to verify our theory.

Given a finite group G, the equitable graph of type II defined on G is an undirected graph with vertex set G, in which two distinct vertices a and b are adjacent if and only if either 0 < | o(a) − o(b)| ≤ min{o(a), o(b)} or one of a and b is equal to e, where o(a) and o(b) are the orders of a and b, respectively. In this paper, we observe that every equitable graph of type II of a finite group is a generalized lexicographic product and discuss finite groups G whose equitable graph of type II is C_n-free, where n ≥ 3. As an application, we show that every equitable graph of type II of a finite group is perfect. Finally, we characterize the metric dimension of the equitable graph of type II of a finite group.