Eccentricity-Based Bounds for the Spectral Radius of Graph Matrices

This article presents a framework for bounding the spectral radius of classical graph matrices, specifically the adjacency and signless Laplacian matrices, using vertex eccentricity as a global structural parameter. By applying the Collatz-Wielandt characterization and Rayleigh quotient methods with the eccentricity vector as a test input, the study derives both upper and lower bounds that incorporate global distance distributions. The proposed bounds are shown to be sharp for regular and self-centered graph families and remain useful for irregular structures. Through these results, eccentricity emerges as a complementary control parameter to degree-based approaches, providing enhanced insight into how both local and global structures shape spectral behaviour.