Signless Laplacian eigenvalue problems of Nordhaus-Gaddum type

2019/04/30 by Xueyi Huang, Huiqiu Lin, Huang, Xueyi +1
Chemistry · Mathematics · #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Synthesis and Properties of Aromatic Compounds

paper · pdf · doi:10.48550/arxiv.1904.13225

openalex publication_date 2019/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a graph of order n, and let q₁(G)≥ q₂(G)≥⋯≥ q_n(G) denote the signless Laplacian eigenvalues of G. Ashraf and Tayfeh-Rezaie [Electron. J. Combin. 21 (3) (2014) #P3.6] showed that q₁(G)+q₁(G)≤ 3n-4, with equality holding if and only if G or G is the star K_1,n-1. In this paper, we discuss the following problem: for n≥6, does q₂(G)+q₂(G)≤ 2n-5 always hold? We provide positive answers to this problem for the graphs with disconnected complements and the bipartite graphs, and determine the graphs attaining the bound. Moreover, we show that q₂(G)+q₂(G)≥ n-2, and the extremal graphs are also characterized.

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