Geodesic
Remarks on spectral radius and Laplacian eigenvalues of a graph
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 781-790
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Let $G$ be a graph with $n$ vertices, $m$ edges and a vertex degree sequence $(d_1, d_2, \dots , d_n)$, where $d_1 \ge d_2 \ge \dots \ge d_n$. The spectral radius and the largest Laplacian eigenvalue are denoted by $\rho (G)$ and $\mu (G)$, respectively. We determine the graphs with \[ \rho (G) = \frac{d_n - 1}{2} + \sqrt{2m - nd_n + \frac{(d_n +1)^2}{4}} \] and the graphs with $d_n\ge 1$ and \[ \mu (G) = d_n + \frac{1}{2} + \sqrt {\sum _{i=1}^n d_i (d_i-d_n) + \Bigl (d_n - \frac{1}{2} \Bigr )^2}. \] We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.
Let $G$ be a graph with $n$ vertices, $m$ edges and a vertex degree sequence $(d_1, d_2, \dots , d_n)$, where $d_1 \ge d_2 \ge \dots \ge d_n$. The spectral radius and the largest Laplacian eigenvalue are denoted by $\rho (G)$ and $\mu (G)$, respectively. We determine the graphs with \[ \rho (G) = \frac{d_n - 1}{2} + \sqrt{2m - nd_n + \frac{(d_n +1)^2}{4}} \] and the graphs with $d_n\ge 1$ and \[ \mu (G) = d_n + \frac{1}{2} + \sqrt {\sum _{i=1}^n d_i (d_i-d_n) + \Bigl (d_n - \frac{1}{2} \Bigr )^2}. \] We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.
Classification : 05C07, 05C50, 05C75
Keywords: spectral radius; Laplacian eigenvalue; strongly regular graph 
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Zhou, Bo; Cho, Han Hyuk. Remarks on spectral radius and Laplacian eigenvalues of a graph. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 781-790. http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a18/

[1] R. A. Brualdi: From the Editor in Chief. Linear Algebra Appl. 360 (2003), 279–283. | MR

[2] D.  M.  Cvetkovič, M.  Doob and H.  Sachs: Spectra of Graphs. DVW, Berlin, 1980. | MR

[3] M.  Fiedler: Algebraic conectivity of graphs. Czechoslovak Math.  J. 23 (1973), 298–305. | MR

[4] R.  Grone and R.  Merris: The Laplacian spectrum of a graph  (II). SIAM J.  Discrete Math. 7 (1994), 221–229. | DOI | MR

[5] Y.  Hong: Bounds of eigenvalues of graphs. Discrete Math. 123 (1993), 65–74. | DOI | MR | Zbl

[6] Y.  Hong, J.  Shu and K.  Fang: A sharp upper bound of the spectral radius of graphs. J.  Combinatorial Theory Ser.  B 81 (2001), 177–183. | DOI | MR

[7] R.  Merris: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197-198 (1994), 143–176. | MR | Zbl

[8] R.  Stanley: A bound on the spectral radius of graphs with $e$  edges. Linear Algebra Appl. 87 (1987), 267–269. | DOI | MR | Zbl

[9] J.  Shu, Y.  Hong and W.  Kai: A sharp upper bound on the largest eigenvalue of the Laplacian matrix of a graph. Linear Algebra Appl. 347 (2002), 123–129. | MR

[10] V.  Nikiforov: Some inequalities for the largest eigenvalue of a graph. Combin. Probab. Comput. 11 (2002), 179–189. | DOI | MR | Zbl

[11] X.  Zhang and J.  Li: On the $k$-th largest eigenvalue of the Laplacian matrix of a graph. Acta Mathematicae Applicatae Sinica 17 (2001), 183–190. | DOI | MR