On distance Laplacian energy in terms of graph invariants

Hilal A. Ganie; Rezwan Ul Shaban; Bilal A. Rather; Shariefuddin Pirzada

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 2, page 335-353
  • ISSN: 0011-4642

Abstract

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For a simple connected graph G of order n having distance Laplacian eigenvalues ρ 1 L ≥ ρ 2 L ≥ ⋯ ≥ ρ n L , the distance Laplacian energy DLE ( G ) is defined as DLE ( G ) = ∑ i = 1 n | ρ i L - 2 W ( G ) / n | , where W ( G ) is the Wiener index of G . We obtain a relationship between the Laplacian energy and the distance Laplacian energy for graphs with diameter 2. We obtain lower bounds for the distance Laplacian energy DLE ( G ) in terms of the order n , the Wiener index W ( G ) , the independence number, the vertex connectivity number and other given parameters. We characterize the extremal graphs attaining these bounds. We show that the complete bipartite graph has the minimum distance Laplacian energy among all connected bipartite graphs and the complete split graph has the minimum distance Laplacian energy among all connected graphs with a given independence number. Further, we obtain the distance Laplacian spectrum of the join of a graph with the union of two other graphs. We show that the graph K k ▽ ( K t ∪ K n - k - t ) , 1 ≤ t ≤ ⌊ n - k 2 ⌋ , has the minimum distance Laplacian energy among all connected graphs with vertex connectivity k . We conclude this paper with a discussion on the trace norm of a matrix and the importance of our results in the theory of the trace norm of the matrix D L ( G ) - ( 2 W ( G ) / n ) I n .

How to cite

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Ganie, Hilal A., et al. "On distance Laplacian energy in terms of graph invariants." Czechoslovak Mathematical Journal 73.2 (2023): 335-353. <http://eudml.org/doc/299360>.

@article{Ganie2023,
abstract = {For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues $ \rho ^\{L\}_\{1\}\ge \rho ^\{L\}_\{2\}\ge \cdots \ge \rho ^\{L\}_\{n\}$, the distance Laplacian energy $\{\rm DLE\} (G)$ is defined as $\{\rm DLE\} (G)=\sum _\{i=1\}^\{n\}|\rho ^\{L\}_i-\{2W(G)\}/\{n\}|$, where $W(G)$ is the Wiener index of $G$. We obtain a relationship between the Laplacian energy and the distance Laplacian energy for graphs with diameter 2. We obtain lower bounds for the distance Laplacian energy $\{\rm DLE\} (G)$ in terms of the order $n$, the Wiener index $W(G)$, the independence number, the vertex connectivity number and other given parameters. We characterize the extremal graphs attaining these bounds. We show that the complete bipartite graph has the minimum distance Laplacian energy among all connected bipartite graphs and the complete split graph has the minimum distance Laplacian energy among all connected graphs with a given independence number. Further, we obtain the distance Laplacian spectrum of the join of a graph with the union of two other graphs. We show that the graph $K_\{k\}\bigtriangledown (K_\{t\}\cup K_\{n-k-t\})$, $1\le t \le \lfloor \frac\{n-k\}\{2\}\rfloor $, has the minimum distance Laplacian energy among all connected graphs with vertex connectivity $k$. We conclude this paper with a discussion on the trace norm of a matrix and the importance of our results in the theory of the trace norm of the matrix $D^L(G)-(2W(G)/n)I_n$.},
author = {Ganie, Hilal A., Ul Shaban, Rezwan, Rather, Bilal A., Pirzada, Shariefuddin},
journal = {Czechoslovak Mathematical Journal},
keywords = {distance matrix; energy; distance Laplacian matrix; distance Laplacian energy},
language = {eng},
number = {2},
pages = {335-353},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On distance Laplacian energy in terms of graph invariants},
url = {http://eudml.org/doc/299360},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Ganie, Hilal A.
AU - Ul Shaban, Rezwan
AU - Rather, Bilal A.
AU - Pirzada, Shariefuddin
TI - On distance Laplacian energy in terms of graph invariants
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 2
SP - 335
EP - 353
AB - For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues $ \rho ^{L}_{1}\ge \rho ^{L}_{2}\ge \cdots \ge \rho ^{L}_{n}$, the distance Laplacian energy ${\rm DLE} (G)$ is defined as ${\rm DLE} (G)=\sum _{i=1}^{n}|\rho ^{L}_i-{2W(G)}/{n}|$, where $W(G)$ is the Wiener index of $G$. We obtain a relationship between the Laplacian energy and the distance Laplacian energy for graphs with diameter 2. We obtain lower bounds for the distance Laplacian energy ${\rm DLE} (G)$ in terms of the order $n$, the Wiener index $W(G)$, the independence number, the vertex connectivity number and other given parameters. We characterize the extremal graphs attaining these bounds. We show that the complete bipartite graph has the minimum distance Laplacian energy among all connected bipartite graphs and the complete split graph has the minimum distance Laplacian energy among all connected graphs with a given independence number. Further, we obtain the distance Laplacian spectrum of the join of a graph with the union of two other graphs. We show that the graph $K_{k}\bigtriangledown (K_{t}\cup K_{n-k-t})$, $1\le t \le \lfloor \frac{n-k}{2}\rfloor $, has the minimum distance Laplacian energy among all connected graphs with vertex connectivity $k$. We conclude this paper with a discussion on the trace norm of a matrix and the importance of our results in the theory of the trace norm of the matrix $D^L(G)-(2W(G)/n)I_n$.
LA - eng
KW - distance matrix; energy; distance Laplacian matrix; distance Laplacian energy
UR - http://eudml.org/doc/299360
ER -

References

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  1. Aouchiche, M., Hansen, P., 10.1016/j.laa.2013.02.030, Linear Algebra Appl. 439 (2013), 21-33. (2013) Zbl1282.05086MR3045220DOI10.1016/j.laa.2013.02.030
  2. Aouchiche, M., Hansen, P., 10.1016/j.laa.2014.06.010, Linear Algebra Appl. 458 (2014), 301-386. (2014) Zbl1295.05093MR3231823DOI10.1016/j.laa.2014.06.010
  3. Aouchiche, M., Hansen, P., 10.1007/s10587-014-0129-2, Czech. Math. J. 64 (2014), 751-761. (2014) Zbl1349.05083MR3298557DOI10.1007/s10587-014-0129-2
  4. Brouwer, A. E., Haemers, W. H., 10.1007/978-1-4614-1939-6, Universitext. Berlin: Springer (2012). (2012) Zbl1231.05001MR2882891DOI10.1007/978-1-4614-1939-6
  5. Cvetković, D. M., Doob, M., Sachs, H., Spectra of Graphs: Theory and Application, Pure and Applied Mathematics 87. Academic Press, New York (1980). (1980) Zbl0824.05046MR0572262
  6. Das, K. C., Aouchiche, M., Hansen, P., 10.1016/j.dam.2018.01.004, Discrete Appl. Math. 243 (2018), 172-185. (2018) Zbl1387.05147MR3804748DOI10.1016/j.dam.2018.01.004
  7. Díaz, R. C., Rojo, O., 10.1016/j.laa.2018.01.032, Linear Algebra Appl. 545 (2018), 55-75. (2018) Zbl1390.05124MR3769113DOI10.1016/j.laa.2018.01.032
  8. Ganie, H. A., 10.1142/S1793830920500615, Discrete Math. Algorithms Appl. 12 (2020), Article ID 2050061, 16 pages. (2020) Zbl1457.05064MR4157019DOI10.1142/S1793830920500615
  9. Ganie, H. A., 10.1016/j.amc.2020.125762, Appl. Math. Comput. 394 (2021), Article ID 125762, 10 pages. (2021) Zbl1462.05222MR4182919DOI10.1016/j.amc.2020.125762
  10. Ganie, H. A., Chat, B. A., Pirzada, S., 10.1016/j.laa.2018.01.021, Linear Algebra Appl. 544 (2018), 306-324. (2018) Zbl1388.05114MR3765789DOI10.1016/j.laa.2018.01.021
  11. Ganie, H. A., Pirzada, S., Rather, B. A., Trevisan, V., 10.1016/j.laa.2019.11.020, Linear Algebra Appl. 588 (2020), 1-18. (2020) Zbl1437.05139MR4037607DOI10.1016/j.laa.2019.11.020
  12. Gutman, I., Zhou, B., 10.1016/j.laa.2005.09.008, Linear Algebra Appl. 414 (2006), 29-37. (2006) Zbl1092.05045MR2209232DOI10.1016/j.laa.2005.09.008
  13. Indulal, G., Gutman, I., Vijayakumar, A., On distance energy of graphs, MATCH Commun. Math. Comput. Chem. 60 (2008), 461-472. (2008) Zbl1199.05226MR2457864
  14. Li, X., Shi, Y., Gutman, I., 10.1007/978-1-4614-4220-2, Springer, New York (2012). (2012) Zbl1262.05100MR2953171DOI10.1007/978-1-4614-4220-2
  15. Monsalve, J., Rada, J., 10.1080/03081087.2018.1448051, Linear Multilinear Algebra 67 (2019), 1121-1131. (2019) Zbl1411.05172MR3937031DOI10.1080/03081087.2018.1448051
  16. Pirzada, S., An Introduction to Graph Theory, Orient Blackswan, Hyderabad (2012). (2012) 
  17. Pirzada, S., Ganie, H. A., 10.1016/j.laa.2015.08.032, Linear Algebra Appl. 486 (2015), 454-468. (2015) Zbl1327.05157MR3401774DOI10.1016/j.laa.2015.08.032
  18. Yang, J., You, L., Gutman, I., Bounds on the distance Laplacian energy of graphs, Kragujevac J. Math. 37 (2013), 245-255. (2013) Zbl1299.05236MR3150862

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