Taiwanese Journal of Mathematics


Ying Liu and Jian Shen

Full-text: Open access


A signed graph $\Gamma=(G, \sigma)$ consists of an unsigned graph $G=(V, E)$ and a mapping $\sigma: E \rightarrow \{+, -\}$. Let $\Gamma$ be a connected signed graph and $L(\Gamma), {\cal L}(\Gamma)$ be its Laplacian matrix and normalized Laplacian matrix, respectively. Suppose $\mu_1\geq \cdots \geq \mu_{n-1}\geq \mu_n\geq 0$ and $\lambda_1\geq \cdots \geq \lambda_{n-1}\geq \lambda_n\geq 0$ are the Laplacian eigenvalues and the normalized Laplacian eigenvalues of $\Gamma$, respectively. In this paper, we give two new lower bounds on $\lambda_1$ which are both stronger than Li's bound [8] and obtain a new upper bound on $\lambda_n$ which is also stronger than Li's bound [8]. In addtion, Hou [6] proposed a conjecture for a connected signed graph $\Gamma: \sum\limits_{i=1}^k\mu _i\gt \sum\limits_{i=1}^k d _i (1\leq k\leq n-1)$. We investigate $\sum\limits_{i=1}^k\mu_i (1\leq k\leq n-1)$ and partly solve the conjecture.

Article information

Taiwanese J. Math., Volume 19, Number 2 (2015), 505-517.

First available in Project Euclid: 4 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

signed graph Laplacian eigenvalues normalized Laplacian eigenvalues


Liu, Ying; Shen, Jian. THE (NORMALIZED) LAPLACIAN EIGENVALUE OF SIGNED GRAPHS. Taiwanese J. Math. 19 (2015), no. 2, 505--517. doi:10.11650/tjm.19.2015.4675. https://projecteuclid.org/euclid.twjm/1499133643

Export citation


  • P. J. Cameron, J. Seidel and J. J. Tsaranov, Signed graphs, root lattices and coxeter groups, J. Algebra, 164(1) (1994), 173-209.
  • F. R. K. Chung, Spectral Graph Theory, CBMS Lecture Notes, Providence, 1997.
  • R. D. Grone, Eigenspaces and the degree sequences of graphs, Linear and Multilinear Algebra, 39 (1995), 133-136.
  • W. H. Haemers, Interlacing Eigenvalues and Graphs, Published in a special issue of Linear Algebra and Its Application in the honour of J. J. Seidel.
  • F. Harary, On the notion of balanced in a signed graph, Michigan Math. J., 2(1) (2012), 143-146.
  • Y. P. Hou, J. S. Li and Y. L. Pan, On the Laplacian eigenvalues of sigened graphs, Linear and Multilinear Algebra, 51(1) (2003), 21-30.
  • H. H. Li, J. S. Li and Y. Z. Fan, The effect on the second smallest eigenvalue of the normalized laplacian of a graph by grafting edges, Linear and Multilinear Algebra, 56(6) (2008), 627-638.
  • H. H. Li and J. S. Li, Note on the normalized Laplacian eigenvalues of signed graphs, Australasian Journal of Combinatorics, 44 (2009), 153-162.
  • H. H. Li and J. S. Li, A note on the normalized laplacian spectra, Taiwanese Journal of Mathematics, 15(1) (2011), 129-139.