Bounds for the largest eigenvalue and sum of Laplacian eigenvalues of signed graphs
Linfeng Xie, Xiaogang Liu
TL;DR
This work develops sharp spectral bounds for signed graphs. It introduces a variational method to bound the largest adjacency eigenvalue in terms of balanced clique structure, and shows that a large spectral radius forces a balanced triangle with a single exceptional case. It then confirms a longstanding conjecture by Hou et al. that, for connected signed graphs, sums of the top Laplacian eigenvalues exceed corresponding sums of top degrees, and provides a general lower bound for these sums based on a chosen vertex subset. Together, the results refine existing bounds and deepen understanding of how balance and topology influence the spectra of signed graphs.
Abstract
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 $Γ$ of order $n$ and size $m$ contains a balanced triangle if $λ_{1}(Γ)\ge \sqrt{m-1}$, $λ_{1}(Γ) \ge |λ_{n}(Γ)|$ and $Γ\not \sim (C_{5}\cup (n-5)K_{1},+)$, where $λ_{1}(Γ)$ is the largest eigenvalue of the adjacency matrix of $Γ$. Thirdly, we confirm a conjecture proposed in [Linear Multilinear Algebra 51 (1) (2003) 21--30] that: if $Γ$ is a connected signed graph, then $$ \sum_{i=1}^{k}μ_{i}(Γ) >\sum_{i=1}^{k}d_{i}(Γ)~~(1\le k\le n-1), $$ where $μ_{1}(Γ)\geμ_{2}(Γ)\ge\cdots \ge μ_{n}(Γ)$ are Laplacian eigenvalues of $Γ$, and $d_{1}(Γ)\ge d_{2}(Γ)\ge \dots \ge d_{n}(Γ)$ are vertex degrees of $Γ$. Finally, we give a lower bound for the sum of the $k$ largest Laplacian eigenvalues of a connected signed graph.