Taiwanese Journal of Mathematics

THE (NORMALIZED) LAPLACIAN EIGENVALUE OF SIGNED GRAPHS

Abstract

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

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

Dates
First available in Project Euclid: 4 July 2017

https://projecteuclid.org/euclid.twjm/1499133643

Digital Object Identifier
doi:10.11650/tjm.19.2015.4675

Mathematical Reviews number (MathSciNet)
MR3332310

Zentralblatt MATH identifier
1357.05087

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

Citation

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

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