Spectra of network related graphs have numerous applications in computer sciences, electrical networks and complex networks to explore structural characterization like stability and strength of these different real-world networks. In present article, our consideration is to compute spectrum based results of generalized prism graph which is well-known planar and polyhedral graph family belongs to the generalized Petersen graphs. Then obtained results are applied to compute some network related quantities like global mean-first passage time, average path length, number of spanning trees, graph energies and spectral radius.

Effective disease control measures are essential for mitigating epidemic risks. This study introduces a novel stochastic susceptible-infected-vaccinated-recovered $\mathcal{S}\mathcal{I}\mathcal{V}\mathcal{R}$ epidemic model that incorporates white noise in vaccination dynamics. Unlike traditional deterministic models, our stochastic framework accounts for the inherent randomness in real-world disease transmission and the effectiveness of interventions. We rigorously establish the existence and uniqueness of global positive solutions using Lyapunov functions and derive conditions for disease extinction and persistence under stochastic perturbations. A key contribution is the introduction of a stochastic reproduction number $R_0^{\ast }$, which refines classical epidemic thresholds by integrating randomness. Through numerical simulations, we illustrate the impact of stochasticity on disease dynamics, demonstrating that noise can drive disease extinction even in scenarios where deterministic models predict persistence. This study provides a more realistic epidemiological framework for optimizing vaccination strategies under uncertainty, offering significant advances in epidemic modeling and public health policy.

This study introduced a novel $\mathcal{S}\mathcal{V}\mathcal{I}\mathcal{R}$ epidemic model incorporating environmental white noise to account for stochastic fluctuations in disease transmission. The model was analyzed to determine conditions for disease persistence and extinction, with outcomes linked to the basic reproduction number. A numerical approach was employed to facilitate computational analysis, and simulations were conducted using data from existing literature to generate realistic predictions. The stochastic model was further evaluated against its deterministic counterpart to assess predictive accuracy. The results highlight the significant role of randomness in epidemiological dynamics, providing valuable insights into disease spread and control strategies.

This investigation focuses on solitary wave solutions and dynamic analysis of the complex coupled Maccari system. We employ the new extended hyperbolic function method to establish bright-wave and dark-wave profiles of the model. The resulting solutions include hyperbolic, trigonometric, and exponential-type functions. Furthermore, we explore the model's dynamical characteristics via multiple perspectives, including phase portrait analysis, quasi-periodic and chaotic patterns, sensitivity analysis and Lyapunov exponent. The analysis validates the robustness of the new extended hyperbolic function method on one hand and extends the understanding of complex wave structures in Maccari's system on the other hand.

In this paper, we study the boundedness of the commutator of the rough fractional Hausdorff operator on grand-variable-Herz-Morrey spaces when the symbol functions belong to bounded mean oscillations (BMO) space.