Remarks on spectral radius and Laplacian eigenvalues of a graph

Bo Zhou; Han Hyuk Cho

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 3, page 781-790
  • ISSN: 0011-4642

Abstract

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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 ρ ( G ) = d n - 1 2 + 2 m - n d n + ( d n + 1 ) 2 4 and the graphs with d n ≥ 1 and μ ( G ) = d n + 1 2 + ∑ i = 1 n d i ( d i - d n ) + d n - 1 2 2 . We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.

How to cite

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Zhou, Bo, and Cho, Han Hyuk. "Remarks on spectral radius and Laplacian eigenvalues of a graph." Czechoslovak Mathematical Journal 55.3 (2005): 781-790. <http://eudml.org/doc/30987>.

@article{Zhou2005,
abstract = {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.},
author = {Zhou, Bo, Cho, Han Hyuk},
journal = {Czechoslovak Mathematical Journal},
keywords = {spectral radius; Laplacian eigenvalue; strongly regular graph; spectral radius; Laplacian eigenvalue; strongly regular graph},
language = {eng},
number = {3},
pages = {781-790},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Remarks on spectral radius and Laplacian eigenvalues of a graph},
url = {http://eudml.org/doc/30987},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Zhou, Bo
AU - Cho, Han Hyuk
TI - Remarks on spectral radius and Laplacian eigenvalues of a graph
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 3
SP - 781
EP - 790
AB - 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.
LA - eng
KW - spectral radius; Laplacian eigenvalue; strongly regular graph; spectral radius; Laplacian eigenvalue; strongly regular graph
UR - http://eudml.org/doc/30987
ER -

References

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  1. From the Editor in Chief, Linear Algebra Appl. 360 (2003), 279–283. (2003) MR1728495
  2. Spectra of Graphs, DVW, Berlin, 1980. (1980) MR0572262
  3. Algebraic conectivity of graphs, Czechoslovak Math.  J. 23 (1973), 298–305. (1973) MR0318007
  4. 10.1137/S0895480191222653, SIAM J.  Discrete Math. 7 (1994), 221–229. (1994) MR1271994DOI10.1137/S0895480191222653
  5. 10.1016/0012-365X(93)90007-G, Discrete Math. 123 (1993), 65–74. (1993) Zbl0788.05067MR1256082DOI10.1016/0012-365X(93)90007-G
  6. 10.1006/jctb.2000.1997, J.  Combinatorial Theory Ser.  B 81 (2001), 177–183. (2001) MR1814902DOI10.1006/jctb.2000.1997
  7. Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197-198 (1994), 143–176. (1994) Zbl0802.05053MR1275613
  8. 10.1016/0024-3795(87)90172-8, Linear Algebra Appl. 87 (1987), 267–269. (1987) Zbl0617.05045MR0878683DOI10.1016/0024-3795(87)90172-8
  9. A sharp upper bound on the largest eigenvalue of the Laplacian matrix of a graph, Linear Algebra Appl. 347 (2002), 123–129. (2002) MR1899886
  10. 10.1017/S0963548301004928, Combin. Probab. Comput. 11 (2002), 179–189. (2002) Zbl1005.05029MR1888908DOI10.1017/S0963548301004928
  11. 10.1007/BF02669571, Acta Mathematicae Applicatae Sinica 17 (2001), 183–190. (2001) MR1877099DOI10.1007/BF02669571

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