The Spectral Radius and the Maximum Degree of Irregular Graphs

Source abstract

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that $$\Delta-\lambda_1>{1\over nD},$$ where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of adding or removing few edges on the spectral radius of a regular graph.