DMGT

Discussiones Mathematicae Graph Theory 28(2) (2008) 345-359
DOI: https://doi.org/10.7151/dmgt.1410

AN UPPER BOUND ON THE LAPLACIAN SPECTRAL RADIUS OF THE SIGNED GRAPHS

Hong-Hai Li

College of Mathematic and Information Science
Jiangxi Normal University Nanchang
JiangXi, 330022 People's Republic of China
e-mail: lhh@mail.ustc.edu.cn

Jiong-Sheng Li

Department of Mathematics
University of Science and Technology of China
Anhui, Hefei 230026 People's Republic of China
e-mail: lijs@ustc.edu.cn

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.

Keywords: Laplacian matrix, signed graph, mixed graph, largest Laplacian eigenvalue, upper bound.

2000 Mathematics Subject Classification: 05C50, 15A18.

References

[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.
[2]	F. Chung, Spectral Graph Theory (CMBS Lecture Notes. AMS Publication, 1997).
[3]	D.M. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs (Academic Press, New York, 1980).
[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.
[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]	R.A. Horn and C.R. Johnson, Matrix Analysis (Cambridge Univ. Press, 1985).
[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.
[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.
[9]	R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197 (1994) 143-176, doi: 10.1016/0024-3795(94)90486-3.
[10]	T.F. Wang, Several sharp upper bounds for the largest Laplacian eigenvalues of a graph, to appear.
[11]	T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982) 47-74, doi: 10.1016/0166-218X(82)90033-6.
[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.
[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.

Received 15 January 2008
Revised 21 April 2008
Accepted 21 April 2008