An upper bound on the Laplacian spectral radius of the signed graphs

Hong-Hai Li; Jiong-Sheng Li

Discussiones Mathematicae Graph Theory (2008)

  • Volume: 28, Issue: 2, page 345-359
  • ISSN: 2083-5892

Abstract

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In this paper, we established a connection between the Laplacian eigenvalues of a signed graph and those of a mixed graph, gave a new upper bound for the largest Laplacian eigenvalue of a signed graph and characterized the extremal graph whose largest Laplacian eigenvalue achieved the upper bound. In addition, an example showed that the upper bound is the best in known upper bounds for some cases.

How to cite

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Hong-Hai Li, and Jiong-Sheng Li. "An upper bound on the Laplacian spectral radius of the signed graphs." Discussiones Mathematicae Graph Theory 28.2 (2008): 345-359. <http://eudml.org/doc/270323>.

@article{Hong2008,
abstract = {In this paper, we established a connection between the Laplacian eigenvalues of a signed graph and those of a mixed graph, gave a new upper bound for the largest Laplacian eigenvalue of a signed graph and characterized the extremal graph whose largest Laplacian eigenvalue achieved the upper bound. In addition, an example showed that the upper bound is the best in known upper bounds for some cases.},
author = {Hong-Hai Li, Jiong-Sheng Li},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {Laplacian matrix; signed graph; mixed graph; largest Laplacian eigenvalue; upper bound},
language = {eng},
number = {2},
pages = {345-359},
title = {An upper bound on the Laplacian spectral radius of the signed graphs},
url = {http://eudml.org/doc/270323},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Hong-Hai Li
AU - Jiong-Sheng Li
TI - An upper bound on the Laplacian spectral radius of the signed graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2008
VL - 28
IS - 2
SP - 345
EP - 359
AB - In this paper, we established a connection between the Laplacian eigenvalues of a signed graph and those of a mixed graph, gave a new upper bound for the largest Laplacian eigenvalue of a signed graph and characterized the extremal graph whose largest Laplacian eigenvalue achieved the upper bound. In addition, an example showed that the upper bound is the best in known upper bounds for some cases.
LA - eng
KW - Laplacian matrix; signed graph; mixed graph; largest Laplacian eigenvalue; upper bound
UR - http://eudml.org/doc/270323
ER -

References

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  1. [1] R.B. Bapat, J.W. Grossman and D.M. Kulkarni, Generalized matrix tree theorem for mixed graphs, Linear and Multilinear Algebra 46 (1999) 299-312, doi: 10.1080/03081089908818623. Zbl0940.05042
  2. [2] F. Chung, Spectral Graph Theory (CMBS Lecture Notes. AMS Publication, 1997). 
  3. [3] D.M. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs (Academic Press, New York, 1980). Zbl0458.05042
  4. [4] J.M. Guo, A new upper bound for the Laplacian spectral radius of graphs, Linear Algebra Appl. 400 (2005) 61-66, doi: 10.1016/j.laa.2004.10.022. Zbl1062.05091
  5. [5] F. Harary, On the notion of balanced in a signed graph, Michigan Math. J. 2 (1953) 143-146, doi: 10.1307/mmj/1028989917. 
  6. [6] R.A. Horn and C.R. Johnson, Matrix Analysis (Cambridge Univ. Press, 1985). Zbl0576.15001
  7. [7] Y.P. Hou, J.S. Li and Y.L. Pan, On the Laplacian eigenvalues of signed graphs, Linear and Multilinear Algebra 51 (2003) 21-30, doi: 10.1080/0308108031000053611. Zbl1020.05044
  8. [8] L.L. Li, A simplified Brauer's theorem on matrix eigenvalues, Appl. Math. J. Chinese Univ. (B) 14 (1999) 259-264, doi: 10.1007/s11766-999-0034-x. Zbl0935.15018
  9. [9] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197 (1994) 143-176, doi: 10.1016/0024-3795(94)90486-3. Zbl0802.05053
  10. [10] T.F. Wang, Several sharp upper bounds for the largest Laplacian eigenvalues of a graph, to appear. 
  11. [11] T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982) 47-74, doi: 10.1016/0166-218X(82)90033-6. Zbl0476.05080
  12. [12] X.D. Zhang, Two sharp upper bounds for the Laplacian eigenvalues, Linear Algebra Appl. 376 (2004) 207-213, doi: 10.1016/S0024-3795(03)00644-X. Zbl1037.05032
  13. [13] X.D. Zhang and J.S. Li, The Laplacian spectrum of a mixed graph, Linear Algebra Appl. 353 (2002) 11-20, doi: 10.1016/S0024-3795(01)00538-9. Zbl1003.05073

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