A Note on Spectral Radius and Maximum Degree of Irregular Graphs

Let G be an irregular graph on n vertices with maximum degree $$\Delta \ge 3$$ and diameter $$D\ge 3$$ . The spectral radius of G, which is denoted by $$\rho (G)$$ , is the largest eigenvalue of the adjacency matrix of G. In this paper, a new lower bound of $$\Delta -\rho (G)$$ is given, which improves the previous bounds intrinsically.