Sharp upper bounds of the spectral radius of a graph

Let G be a simple connected graph with n vertices and m edges. The spectral radius ρ ( G ) of G is the largest eigenvalue of its adjacency matrix. In this paper, we firstly consider the effect on the spectral radius of a graph by removing a vertex, and then as an application of the result, we obtain a new sharp upper bound of ρ ( G ) which improves some known bounds: If ( k − 2 ) ( k − 3 ) 2 ≤ m − n ≤ k ( k − 3 ) 2 , where k ( 3 ≤ k ≤ n ) is an integer, then ρ ( G ) ≤ 2 m − n − k + 5 2 + 2 m − 2 n + 9 4 . The equality holds if and only if G is a complete graph K n or K 4 − e , where K 4 − e is the graph obtained from K 4 by deleting some edge e .