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 $\mathrm{\Gamma }$ of order $n$ and size $m$ contains a balanced triangle if ${\lambda }_1(\mathrm{\Gamma })\ge \sqrt{m-1}$, ${\lambda }_1(\mathrm{\Gamma })\ge \mathrm{\mid }{\lambda }_n(\mathrm{\Gamma })\mathrm{\mid }$ and $\mathrm{\Gamma }\sim ̸(C_5\cup (n-5)K_1,+)$, where ${\lambda }_1(\mathrm{\Gamma })$ is the largest eigenvalue of the adjacency matrix of $\mathrm{\Gamma }$. Thirdly, we confirm a conjecture proposed in [Linear Multilinear Algebra 51 (1) (2003) 21–30] that: if $\mathrm{\Gamma }$ is a connected signed graph, then $$\sum _{i=1}^k{\mu }_i(\mathrm{\Gamma })>\sum _{i=1}^k{d}_i(\mathrm{\Gamma })(1\le k\le n-1),$$ where ${\mu }_1(\mathrm{\Gamma })\ge {\mu }_2(\mathrm{\Gamma })\ge \cdots \ge {\mu }_n(\mathrm{\Gamma })$ are Laplacian eigenvalues of $\mathrm{\Gamma }$, and $d_1(\mathrm{\Gamma })\ge d_2(\mathrm{\Gamma })\ge \cdots \ge d_n(\mathrm{\Gamma })$ are vertex degrees of $\mathrm{\Gamma }$. Finally, we give a lower bound for the sum of the $k$ largest Laplacian eigenvalues of a connected signed graph.