Maximum Sombor Index and Spectral Radius of Hypertrees
In the framework of complex network analysis, hypergraphs provide a natural generalization for modeling higher-order interactions. In this work, we investigate the extremal structural properties of uniform hypertrees with respect to the Sombor index and the Sombor spectral radius, two degree-based measures that capture nonlinear connectivity patterns. For hypertrees of fixed size, we identify the structures that maximize and second-maximize these descriptors. We show that both the Sombor index and the Sombor spectral radius increase strictly under an edge-releasing operation applied to non-pendant hyperedges, revealing a monotonic structural transformation principle. This result enables us to characterize the hyperstar configuration as the unique maximizer of both measures. Furthermore, by systematically employing edge-moving and edge-releasing operations, we determine the hypertree structure that attains the second-highest values of these indices. Our findings contribute to the understanding of how local structural modifications influence global spectral and topological descriptors in higher-order networks, offering insights relevant to the study of nonlinear and complex systems.