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Huang, Xueyi; Lin, Huiqiu
Linear algebra and its applications, 11/2019, Letnik: 581Journal Article
Let G be a graph of order n, and let q1(G)≥q2(G)≥⋯≥qn(G) denote the signless Laplacian eigenvalues of G. Ashraf and Tayfeh-Rezaie (2014) 3 showed that q1(G)+q1(G‾)≤3n−4, with equality holding if and only if G or G‾ is the star K1,n−1. In this paper, we prove that q2(G)+q2(G‾)≤2n−4, where the equality holds if and only if G or G‾ is K2, P4 or C4. Also, 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.
