## Abstract

Let *G* be a connected graph with *n* vertices, *m* edges and having distance signless Laplacian eigenvalues *ρ*_{1}*≥ ρ*_{2} ≥ *… ≥ ρ*_{n}*≥* 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 *α*^{th} powers of the distance signless Laplacian eigenvalues of the graph *G*. In this paper, we obtain various bounds for the graph invariant *m*_{α}(*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*).

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## 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.

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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

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DOI: https://doi.org/10.1007/s13226-020-0455-z