Sharp upper bounds of the spectral radius of a graph

Abstract

Let G be a simple connected graph with n vertices and m edges. The spectral radiusρ(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)≤2m−n−k+52+2m−2n+94.The equality holds if and only if G is a complete graph Kn or K4−e, where K4−e is the graph obtained from K4 by deleting some edge e.