Remarks on Spectral Radius and Laplacian Eigenvalues of a Graph

AbstractLet G be a graph with n vertices, m edges and a vertex degree sequence (d1, d2,..., dn), where d1 ≥ d2 ≥ ... ≥ dn. 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 dn ≥ 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.

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