# A note on the least (normalized) laplacian eighva;ue of signed graphs

• Hui Shu Li
• Hong Hai Li

## Keywords:

signed graph, Laplacian, eigenvalues, balancedness

## Abstract

Let $\Gamma=(G, \sigma)$ be a connected signed graph, and $L(\Gamma)$ be its Laplacian and $\mathcal{L}(\Gamma)$ its normalized Laplacian with eigenvalues $\lambda_1\geq \lambda_2\geq\cdots \geq \lambda_n$ and $\mu_1\geq \mu_2\geq\cdots \geq \mu_n$, respectively. It is known that a signed graph $\Gamma$ is balanced if and only if $\lambda_n=0$ (or $\mu_n=0$). We show that $\lambda_n$ and $\mu_n$ measure how much $\Gamma$ is far from being balanced by proving that \begin{align*}\mu_n(\Gamma)&\leq\min\{\frac{2\epsilon(\Gamma)}{m}, \frac{\nu(\Gamma)}{\nu(\Gamma)+\nu_1(\Gamma)}\},\\ \lambda_n(\Gamma)&\leq \min\{\lambda_1(\Gamma'): \Gamma-\Gamma'\,\,\,{is balanced}\}, \end{align*}where $\nu(\Gamma)$ (resp. $\epsilon(\Gamma)$) denotes the frustration number (resp. the frustration index) of $\Gamma$, that is the minimum number of vertices (resp. edges) to be deleted such that the signed graph is balanced.

## Author Biography

### Hong Hai Li

College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, JiangXi, 330022, People’s Republic of China.

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2016-09-30

## How to Cite

Li, H. S., & Li, H. H. (2016). A note on the least (normalized) laplacian eighva;ue of signed graphs. Tamkang Journal of Mathematics, 47(3), 271-278. https://doi.org/10.5556/j.tkjm.47.2016.1942

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