Spectral Analysis of Epidemic Dynamics via the Kirchhoff Index

This paper develops a spectral framework for analyzing epidemic dynamics on weighted networks through the Kirchhoff index and the Laplacian spectrum. Starting from a nonlinear network-based SIS model, we investigate the disease-free and endemic equilibria and establish their local stability conditions in terms of the spectral radius of the adjacency matrix. We then introduce a spectral criterion based on the Kirchhoff index, deriving bounds that connect the total effective resistance of the network with the algebraic connectivity and the characteristic relaxation time. A dimensionless parameter combining epidemic transmission and network topology is proposed, leading to a critical Kirchhoff threshold that separates highly resilient networks from weakly connected and more sensitive structures. These results are summarized in the Kirchhoff Stability Principle, which provides a unified relationship between the Kirchhoff index, algebraic connectivity, relaxation time, and dynamical robustness. Numerical illustrations on several canonical graph topologies, including complete, star, cycle, and path graphs, validate the theoretical findings and demonstrate that networks with smaller Kirchhoff indices exhibit faster convergence and stronger resilience against perturbations. The proposed framework offers a simple spectral approach for understanding the interplay between graph topology and epidemic dynamics and can be extended to more general epidemic and network models.