A relation between the Laplacian and signless Laplacian eigenvalues of a graph
Saieed Akbari
, Ebrahim Ghorbani
, Jack H. Koolen
and Mohammad Reza Oboudi
DOI: 10.1007/s10801-010-0225-9
Abstract
Let G be a graph of order n such that å i=0 n( -1) i a i l n - i \sum_{i=0}^{n}(-1)^{i}a_{i}λ^{n-i} and å i=0 n( -1) i b i l n - i \sum_{i=0}^{n}(-1)^{i}b_{i}λ^{n-i} are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that a i \geq b i for i=0,1,\cdots , n. As a consequence, we prove that for any α , 0< α \leq 1, if q 1,\cdots , q n and μ 1,\cdots , μ n are the signless Laplacian and the Laplacian eigenvalues of G, respectively, then q 1 a+ \frac{1}{4} + q n a $^{3}$ m 1 a+ \frac{1}{4} + m n a q_{1}^{α}+\cdots+q_{n}^{α}\geq\mu_{1}^{α}+\cdots+μ_{n}^{α}.
Pages: 459–464
Keywords: keywords Laplacian; signless Laplacian; incidence energy; Laplacian-like energy
Full Text: PDF
References
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2. Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1993)
3. Efroymson, G.A., Swartz, B., Wendroff, B.: A new inequality for symmetric functions. Adv. Math. 38, 109-127 (1980)
4. Grone, R., Merris, R., Sunder, V.S.: The Laplacian spectrum of a graph. SIAM J. Matrix Anal. Appl. 11, 218-238 (1990)
5. Gutman, I., Kiani, D., Mirzakhah, M.: On incidence energy of graphs. MATCH Commun. Math. Comput. Chem. 62, 573-580 (2009)
6. Gutman, I., Kiani, D., Mirzakhah, M., Zhou, B.: On incidence energy of a graph. Linear Algebra Appl. 431, 1223-1233 (2009)
7. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
8. Jooyandeh, M.R., Kiani, D., Mirzakhah, M.: Incidence energy of a graph. MATCH Commun. Math. Comput. Chem. 62, 561-572 (2009)
9. Liu, J., Liu, B.: A Laplacian-energy like invariant of a graph. MATCH Commun. Math. Comput.
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