2025/12/01 by Xie, Linfeng, Liu, Xiaogang
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.01736
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 λ1(Γ)≥ √(m-1), λ1(Γ) ≥ |λn(Γ)| and Γ\not ∼ (C5∪ (n-5)K1,+), where λ1(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 ∑i=1kμi(Γ) gt;∑i=1kdi(Γ)~~(1≤ k≤ n-1), where μ1(Γ)≥μ2(Γ)≥⋯ ≥ μn(Γ) are Laplacian eigenvalues of Γ, and d1(Γ)≥ d2(Γ)≥ … ≥ dn(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.