THE (NORMALIZED) LAPLACIAN EIGENVALUE OF SIGNED GRAPHS

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 $\Gamma: \sum\limits_{i=1}^k\mu _i>\sum\limits_{i=1}^k d _i (1\leq k\leq n-1)$. We investigate $\sum\limits_{i=1}^k\mu_i (1\leq k\leq n-1)$ and partly solve the conjecture.