This paper presents a study of scattering from a ring graph, a simple model used to understand complex (chaotic) phenomena in scattering from more complex graphs. The ring graph is shown to exhibit several key features, including the appearance of arbitrarily narrow resonances, known as "topological resonances," which are directly linked to the existence of cycles. These resonances are sensitive to perturbations, such as time-dependent random noise, and their behavior is analyzed in detail. The ring graph consists of two inner bonds forming a circle connected to two leads. The wave functions on the bonds and leads are derived, and the scattering matrix elements and resonances are computed. The transmission and reflection amplitudes are determined, and resonances are identified as poles of the scattering matrix in the complex k-plane. The real parts of these poles determine the wave number where resonances occur, while the imaginary parts are proportional to the resonance widths. The structure of the resonances depends on the ratio of the lengths of the bonds. When the lengths are equal, the zeros of the denominator function D(k) correspond to complex poles, leading to broad resonances. When the lengths are rationally related, bound states appear, which are embedded in the continuum. For irrational ratios, these bound states become resonances with very narrow widths, known as topological resonances. The paper also analyzes the effect of a random, time-dependent perturbation on the ring graph. A delta potential is introduced at a time-dependent position, and the effect on the transmission is studied. The current density is calculated perturbatively, and the results show that the effect of the perturbation increases with the narrowing of the resonance. The average behavior of the noise is studied, and it is shown that the resonance broadening depends on the parameters α and σ². In conclusion, the ring graph is shown to exhibit interesting features such as very narrow resonances, making it a useful model for understanding chaotic scattering on graphs. The study also highlights the importance of considering environmental coupling in quantum systems, as it affects the behavior of resonances and scattering processes.This paper presents a study of scattering from a ring graph, a simple model used to understand complex (chaotic) phenomena in scattering from more complex graphs. The ring graph is shown to exhibit several key features, including the appearance of arbitrarily narrow resonances, known as "topological resonances," which are directly linked to the existence of cycles. These resonances are sensitive to perturbations, such as time-dependent random noise, and their behavior is analyzed in detail. The ring graph consists of two inner bonds forming a circle connected to two leads. The wave functions on the bonds and leads are derived, and the scattering matrix elements and resonances are computed. The transmission and reflection amplitudes are determined, and resonances are identified as poles of the scattering matrix in the complex k-plane. The real parts of these poles determine the wave number where resonances occur, while the imaginary parts are proportional to the resonance widths. The structure of the resonances depends on the ratio of the lengths of the bonds. When the lengths are equal, the zeros of the denominator function D(k) correspond to complex poles, leading to broad resonances. When the lengths are rationally related, bound states appear, which are embedded in the continuum. For irrational ratios, these bound states become resonances with very narrow widths, known as topological resonances. The paper also analyzes the effect of a random, time-dependent perturbation on the ring graph. A delta potential is introduced at a time-dependent position, and the effect on the transmission is studied. The current density is calculated perturbatively, and the results show that the effect of the perturbation increases with the narrowing of the resonance. The average behavior of the noise is studied, and it is shown that the resonance broadening depends on the parameters α and σ². In conclusion, the ring graph is shown to exhibit interesting features such as very narrow resonances, making it a useful model for understanding chaotic scattering on graphs. The study also highlights the importance of considering environmental coupling in quantum systems, as it affects the behavior of resonances and scattering processes.