More inequalities for Laplacian indices by way of majorization

Document Type: Research Paper

Author

The University of New Mexico, Albuquerque, NM 87131, USA

Abstract

The n-tuple of Laplacian characteristic values of a graph is majorized by the conjugate sequence of its degrees. Using that result we find a collection of general inequalities for a number of Laplacian indices expressed in terms of the conjugate degrees, and then with a maximality argument, we find tight general bounds expressed in terms of the size of the vertex set n and the average degree dG = 2|E|/n. We also find some particular tight bounds for some classes of graphs in terms of customary graph parameters.

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Main Subjects


1. H. Bai, The Grone-Merris conjecture, Trans. Amer. Math. Soc. 363 (2011)
4463−4474.
2. M. Bianchi, A. Cornaro, J. L. Palacios, A. Torriero, Localization of graph
topological indices via majorization technique, in: M. Dehmer, F. Emmert-Streib
(Eds.), Quantitative Graph Theory–Mathematical Foundations and Applications,
CRC Press, Boca Raton, 2015, pp. 35–79.
3. A. Cafure, D. A. Jaume, L. N. Grippo, A. Pastine, M. D. Safe, V. Trevisan, I.
Gutman, Some Results for the (Signless) Laplacian Resolvent, MATCH Commun.
Math. Comput. Chem. 77 (2017) 105–114.
4. K. C. Das, K. Su, M. Liu, Sums of powers of eigenvalues of the Laplacian,
Linear Algebra Appl. 439 (2013) 3561–3575.

5. K. C. Das, K. S. A. Mojallal, I. Gutman, Relations between degrees, conjugate
degrees and graph energies, Linear Algebra Appl. 515 (2017) 24–37.
6. M. Eliasi, A simple approach to order the multiplicative Zagreb indices of
connected graphs, Trans. Comb. 1 (2012) 17–24.
7. R. Grone, R. Merris, The Laplacian spectrum of a graph II, Siam J. Discrete
Math. 7 221−229.
8. I. Gutman, B. Mohar, The quasi-Wiener and the Kirchhoff indices coincide, J.
Chem. Inf. Comput. Sci. 36 (1996) 982–985.
9. I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414
(2006) 29–37.
10. I. Gutman, D. Kiani, M. Mirzakhah, B. Zhou, On incidence energy of a graph,
Linear Algebra Appl. 431 (2009) 1223–1233.
11. I. Gutman, B. Furtula, E. Zogić, E. Glogić, Resolvent energy of graphs, MATCH
Commun. Math. Comput. Chem. 75 (2016) 279–290.
12. D. J. Klein, M. Randić, Resistance distance, J. Math. Chem. 12 (1993) 81-95.
13. J. Liu, B. Liu, A Laplacian-energy like invariant of a graph, MATCH Commun.
Math. Comput. Chem. 59 (2008) 355–372.
14. A. W. Marshall, I. Olkin, Inequalities–Theory of Majorization and its
Applications, Academic Press, London, 1979.
15. J. L. Palacios, Some inequalities for Laplacian descriptors via majorization,
MATCH Commun. Math. Comput. Chem. 77 (2017) 189–194.
16. Y. Yang, On a new cyclicity measure of graphs–The global cyclicity index.
Discr.Appl. Math. 172 (2014) 88–97.
17. B. Zhou, On a sum of powers of the Laplacian eigenvalues of a graph, Linear
Alg. Appl. 429 (2008) 2239–2246.
18. H. Y. Zhu, D. J. Klein, I. Lukovits, Extensions of the Wiener number, J. Chem.
Inf.Comput. Sci. 36 (1996) 420–428.