Abstract
Let G be a graph of order n such that \(\sum_{i=0}^{n}(-1)^{i}a_{i}\lambda^{n-i}\) and \(\sum_{i=0}^{n}(-1)^{i}b_{i}\lambda^{n-i}\) are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that a i ≥b i for i=0,1,…,n. As a consequence, we prove that for any α, 0<α≤1, if q 1,…,q n and μ 1,…,μ n are the signless Laplacian and the Laplacian eigenvalues of G, respectively, then \(q_{1}^{\alpha}+\cdots+q_{n}^{\alpha}\geq\mu_{1}^{\alpha}+\cdots+\mu _{n}^{\alpha}\).
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Akbari, S., Ghorbani, E., Koolen, J.H. et al. A relation between the Laplacian and signless Laplacian eigenvalues of a graph. J Algebr Comb 32, 459–464 (2010). https://doi.org/10.1007/s10801-010-0225-9
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DOI: https://doi.org/10.1007/s10801-010-0225-9