Remarks on Spectral Radius and Laplacian Eigenvalues of a Graph

Abstract

Let G be a graph with n vertices, m edges and a vertex degree sequence (d ₁, d ₂,..., d _n ), where d ₁ ≥ d ₂ ≥ ... ≥ d _n . The spectral radius and the largest Laplacian eigenvalue are denoted by ϱ(G) and µ(G), respectively. We determine the graphs with

$$\varrho (G) = \frac{{d_n - 1}}{2} + \sqrt {2m - nd_n + \frac{{(d_n + 1)^2 }}{4}}$$

and the graphs with d _n ≥ 1 and

$$\mu (G) = d_n + \frac{1}{2} + \sqrt {\sum\limits_{i - 1}^n {di(di - dn) + } \left( {d_n - \frac{1}{2}} \right)^2 .}$$

We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.