Matrix analysis
The Laplacian spectrum of a graph
SIAM Journal on Matrix Analysis and Applications
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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 驴b i for i=0,1,驴,n. As a consequence, we prove that for any 驴, 0驴驴1, if q 1,驴,q n and μ 1,驴,μ n are the signless Laplacian and the Laplacian eigenvalues of G, respectively, then $q_{1}^{\alpha}+\cdots+q_{n}^{\alpha}\geq\mu_{1}^{\alpha}+\cdots+\mu _{n}^{\alpha}$ .