On the largest eigenvalue of non-regular graphs

We study the spectral radius of connected non-regular graphs. Let λ1(n,Δ) be the maximum spectral radius among all connected non-regular graphs with n vertices and maximum degree Δ. We prove that Δ−λ1(n,Δ)=Θ(Δ/n2). This improves two recent results by Stevanović and Zhang, respectively.