Abstract
Let G be a connected graph with n vertices, m edges and having distance signless Laplacian eigenvalues ρ1≥ ρ2 ≥ … ≥ ρn≥ 0. For any real number α ≠ 0, let \({m_\alpha }\left( G \right) = \sum\nolimits_{i = 1}^n {\rho _i^\alpha } \) be the sum of αth powers of the distance signless Laplacian eigenvalues of the graph G. In this paper, we obtain various bounds for the graph invariant mα(G), which connects it with different parameters associated to the structure of the graph G. We also obtain various bounds for the quantity DEL(G), the distance signless Laplacian-energy-like invariant of the graph G. These bounds improve some previously known bounds. We also pose some extremal problems about DEL(G).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Akbari, E. Ghorbani, J. H. Koolen, and M. R. Oboudi, On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs, Elec. J. Comb., 17 (2010), R115.
A. Alhevaz, M. Baghipur, S. Pirzada, and Y. Shang, Some bounds for distance signless Laplacian energylike invariant of graphs, Symmetry, to appear.
M. Aouchiche and P. Hansen, Two Laplacians for the distance matrix of a graph, Linear Algebra Appl., 439(1) (2013), 21–33.
M. Aouchiche and P. Hansen, Distance spectra of graphs: A survey, Linear Algebra Appl., 458 (2014), 301–386.
F. Ashraf, On two conjectures on sum of the powers of signless Laplacian eigenvalues of a graph, Linear and Multilinear Algebra, 64 (2016), 1314–1320.
R. B. Bapat, Determinant of the distance matrix of a tree with matrix weights, Linear Algebra Appl., 416 (2006), 2–7.
R. B. Bapat, S. J. Kirkland, and M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl., 401 (2005), 193–209.
R. B. Bapat, A. K. Lal, and S. Pati, A q-analogue of the distance matrix of a tree, Linear Algebra Appl., 416 (2006), 799–814.
K. C. Das, K. Xu, and M. Liu, On sum of powers of the Laplacian eigenvalues of graphs, Linear Algebra Appl., 439 (2013), 3561–3575.
K. C. Das, M. Aouchiche, and P. Hansen, On (distance) Laplacian energy and (distance) signless Laplacian energy of graphs, Discrete Applied Math., 243 (2018), 172–185.
R. C. Diaz and O. Rojo, Sharp upper bounds on the distance energies of a graph, Linear Algebra Appl., 545 (2018), 55–75.
H. A. Ganie, A. M. Alghamdi, and S. Pirzada, On the sum of the Laplacian eigenvalues of a graph and Brouwer’s conjecture, Linear Algebra Appl., 501 (2016), 376–389.
J. Liu and B. Liu, A Laplacian-energy-like invariant of a graph, MATCH Commun. Math. Comput. Chem., 59 (2008), 355–372.
L. Lu, Q. Huang, and X. Huang, On graphs whose smallest distance (signless Laplacian) eigenvalue has large multiplicity, Linear Multilinear Algebra, 66, 11 (2018), 2218–2231.
S. Pirzada, An introduction to graph theory, Universities Press, OrientBlackSwan, Hyderabad, (2012).
S. Pirzada and H. A. Ganie, On Laplacian-energy-like invariant and incidence energy, Trans. Comb., 4 (2015), 41–51.
S. Pirzada, H. A. Ganie, and I. Gutman. On Laplacian-energy-like invariant and Kirchhoff index, MATCH Commun. Math. Comput. Chem., 73 (2015), 41–59.
B. Zhou. On sum of powers of the Laplacian eigenvalues of graphs, Linear Algebra Appl., 429 (2008) 2239–2246.
Acknowledgement
The authors thank the referee for his useful comments and suggestions. The research of S. Pirzada is supported by SERB-DST, New Delhi, under the research project number MTR/2017/000084.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Pirzada, S., Ganie, H.A., Alhevaz, A. et al. On the Sum of the Powers of Distance Signless Laplacian Eigenvalues of Graphs. Indian J Pure Appl Math 51, 1143–1163 (2020). https://doi.org/10.1007/s13226-020-0455-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-020-0455-z