On distance Laplacian spectral invariants of brooms and their complements

Document Type : Original paper

Authors

1 Mathematical Sciences Department, College of Science, United Arab Emirates University, UAE.

2 University of Kashmir

Abstract

For a connected graph $G$ of order $n$, the distance Laplacian matrix $D^L(G)$ is defined as $D^L(G)=Tr(G)-D(G)$, where $Tr(G)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The largest eigenvalue of $D^L(G)$ is the distance Laplacian spectral radius of $G$ and the quantity $DLE(G)=\sum\limits_{i=1}^{n}|\rho^L_i(G)-\frac{2W(G)}{n}|$, where $W(G)$ is the Wiener index of $G$, is the distance Laplacian energy of $G$. Brooms of diameter $4$ are the trees obtained from the path $P_{5}$ by appending pendent vertices at some vertex of $ P_{5}$. One of the interesting and important problems in spectral graph theory is to find extremal graphs for a spectral graph invariant and ordering them according to this graph invariant. This problem has been considered for many families of graphs with respect to different graph matrices. In the present article, we consider this problem for brooms of diameter $4$ and their complements with respect to their distance Laplacian matrix. Formally, we discuss the distance Laplacian spectrum and the distance Laplacian energy of brooms of diameter $4$. We will prove that these families of trees can be ordered in terms of their distance Laplacian energy and the distance Laplacian spectral radius. Further, we obtain the distance Laplacian spectrum and the distance Laplacian energy of complement of the family of double brooms and order them in terms of the smallest non-zero distance Laplacian eigenvalue and the distance Laplacian energy.

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References

[1] M. Ajmal, M.U. Rehman, and T. Kamran, The second least eigenvalue of the signless Laplacian of the complements of trees, Electron. J. Graph Theory Appl. 7 (2019), no. 2, 265–275.
https://dx.doi.org/10.5614/ejgta.2019.7.2.6
[2] A. Alhevaz, M. Baghipur, H. A. Ganie, and Y. Shang, The generalized distance spectrum of the join of graphs, Symmetry 12 (2020), no. 1, Article ID: 169.
https://doi.org/10.3390/sym12010169
[3] A. Alhevaz, S. Pirzada, and Y. Shang, Some inequalities involving the distance signless Laplacian eigenvalues of graphs, Trans. Comb. 10 (2021), no. 1, 9–29.
https://doi.org/10.22108/toc.2020.121940.1715
[4] M. Aouchiche and P. Hansen, Two Laplacians for the distance matrix of a graph, Linear Algebra Appl. 439 (2013), no. 1, 21–33.
https://doi.org/10.1016/j.laa.2013.02.030
[5] M. Aouchiche and P. Hansen, Distance spectra of graphs: A survey, Linear Algebra Appl. 458 (2014), 301–386.
https://doi.org/10.1016/j.laa.2014.06.010
[6] M. Aouchiche and P. Hansen, Some properties of the distance Laplacian eigenvalues of a graph, Czech. Math. J. 64 (2014), no. 3, 751–761.
https://doi.org/10.1007/s10587-014-0129-2
[7] M. Baghipur, M. Ghorbani, H. A Ganie, and Y. Shang, On the second-largest reciprocal distance signless Laplacian eigenvalue, Mathematics 9 (2021), no. 5, Article ID: 512.
https://doi.org/10.3390/math9050512
[8] D. Cvetkovi´c, P. Rowlinson, and S. Simi´c, An Introduction to the Theory of Graph Spectra, London Mathematical Society Student Texts, Cambridge University Press, 2009.
[9] K.C. Das, M. Aouchiche, and P. Hansen, On (distance) Laplacian energy and (distance) signless Laplacian energy of graphs, Discrete Appl. Math. 243 (2018), 172–185.
https://doi.org/10.1016/j.dam.2018.01.004
[10] R.R. Del Vecchio, I. Gutman, V. Trevisan, and C.T.M. Vinagre, On the spectra and energies of double-broom-like trees, Kragujevac J. Sci. 31 (2009), 45–58.
[11] R.C. Díaz and O. Rojo, Sharp upper bounds on the distance energies of a graph, Linear Algebra Appl. 545 (2018), 55–75.
https://doi.org/10.1016/j.laa.2018.01.032
[12] H. Ganie, On distance Laplacian spectrum (energy) of graphs, Discrete Math. Algorithms Appl. 12 (2020), no. 5, Article ID: 2050061
https://doi.org/10.1142/S1793830920500615
[13] H.A. Ganie, On the distance Laplacian energy ordering of a tree, Appl. Math. Comput. 394 (2021), Article ID: 125762.
https://doi.org/10.1016/j.amc.2020.125762
[14] I. Gutman and B. Furtula, Graph energies and their applications, Bull. Cl. Sci. Math. Nat. Sci. Math. (2019), no. 44, 29–45.
[15] R.A. Horn and C.R. Johnson, Matrix Analysis, Second Edition, Cambridge university press, 2012.
[16] S. Li and S. Wang, The least eigenvalue of the signless Laplacian of the complements of trees, Linear Algebra Appl. 436 (2012), no. 7, 2398–2405.
https://doi.org/10.1016/j.laa.2011.09.032
[17] H. Lin and B. Zhou, On the distance Laplacian spectral radius of graphs, Linear Algebra Appl. 475 (2015), 265–275.
https://doi.org/10.1016/j.laa.2015.02.033
[18] M. Nath and S. Paul, On the distance Laplacian spectra of graphs, Linear Algebra Appl. 460 (2014), 97–110.
https://doi.org/10.1016/j.laa.2014.07.025
[19] S. Pirzada, B.A. Rather, and T. A. Chishti, On distance Laplacian spectrum of zero divisor graphs of Zn, Carpathian J. Math. 13 (2021), no. 1, 48–57.
https://doi.org/10.15330/cmp.13.1.48-57
[20] B.A. Rather, H.A. Ganie, and Y. Shang, Distance Laplacian spectral ordering of sun type graphs, Appl. Math. Comput. 445 (2023), Article ID: 127847.
https://doi.org/10.1016/j.amc.2023.127847
[21] V. Trevisan, J.B. Carvalho, R.R. Del Vecchio, and C.T.M. Vinagre, Laplacian energy of diameter 3 trees, Appl. Math. Lett. 24 (2011), no. 6, 918–923.
https://doi.org/10.1016/j.aml.2010.12.050
 
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 18 January 2024

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