A Sharp Upper Bound of the Spectral Radius of Graphs

Let G be a simple connected graph with n vertices and m edges. Let δ(G)=δ be the minimum degree of vertices of G. The spectral radius ρ(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, we obtain the following sharp upper bound of ρ(G): ρ(G)⩽δ−1+(δ+1)2+4(2m−δn)2.Equality holds if and only if G is either a regular graph or a bidegreed graph in which each vertex is of degree either δ or n−1.