Bounds for the largest eigenvalue and sum of Laplacian eigenvalues of signed graphs

In this paper, we consider the bounds for the largest eigenvalue and the sum of the $k$ largest Laplacian eigenvalues of signed graphs. Firstly, we give an upper bound on the largest eigenvalue of the adjacency matrix of a signed graph and characterize the extremal graphs that attain this bound. Secondly, we prove that a non-bipartite signed graph $\Gamma$ of order $n$ and size $m$ contains a balanced triangle if $\lambda_{1}(\Gamma)\ge \sqrt{m-1}$, $\lambda_{1}(\Gamma) \ge |\lambda_{n}(\Gamma)|$ and $\Gamma\not \sim (C_{5}\cup (n-5)K_{1},+)$, where $\lambda_{1}(\Gamma)$ is the largest eigenvalue of the adjacency matrix of $\Gamma$. Thirdly, we confirm a conjecture proposed in [Linear Multilinear Algebra 51 (1) (2003) 21--30] that: if $\Gamma$ is a connected signed graph, then $$\sum_{i=1}^{k}\mu_{i}(\Gamma) >\sum_{i=1}^{k}d_{i}(\Gamma)~~(1\le k\le n-1),$$ where $\mu_{1}(\Gamma)\ge\mu_{2}(\Gamma)\ge\cdots \ge \mu_{n}(\Gamma)$ are Laplacian eigenvalues of $\Gamma$, and $d_{1}(\Gamma)\ge d_{2}(\Gamma)\ge \dots \ge d_{n}(\Gamma)$ are vertex degrees of $\Gamma$. Finally, we give a lower bound for the sum of the $k$ largest Laplacian eigenvalues of a connected signed graph.