Bounds on the spectral radius of graphs via subgraph deletion and coalescence

Abstract

The spectral radius of a graph is a fundamental parameter closelylinked to network dynamics and robustness. Understanding its precisevariation under structural perturbations such as vertex deletion,edge removal, or graph coalescence is a central problem in spectralgraph theory. In this paper, we establish several sharp upper boundsthat quantify these changes with an explicit dependence on localgraph structures. For the deletion of an arbitrary nonempty propervertex set $S$, we give an exact Schur-complement formula for$\rho(G)$ in terms of $G[S]$, $G-S$, and the incidence matrix between them.The formula gives a precise relation between the spectral radius before and after deleting $S$ and yields a sharp upper bound withan equality characterization. In the special case where $S$ is a clique, this gives an explicit clique-deletion estimate. We also derive an alternative local bound for graph coalescence that captures the degrees at the identified root and is sometimes sharperthan the bound of Wang and Guo [(2020) \textit{Linear Algebra Appl.}, 601:101-112, 2020.]. For the deletion of the endpoints of an edge, we obtain a bound that incorporates the number of triangles containing the specified edge. We characterize equality in these upper bounds.