The spectral radius and the maximum degree of irregular graphs

Abstract

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that Δ-λ_1>\frac{1}{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.

10 pages, 1 figure, submitted to EJC on January 20, 2007