Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
distance matrix; energy; distance Laplacian matrix; distance Laplacian energy
distance matrix; energy; distance Laplacian matrix; distance Laplacian energy
Summary:
For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues $ \rho ^{L}_{1}\geq \rho ^{L}_{2}\geq \cdots \geq \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\leq t \leq \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 $D^L(G)-(2W(G)/n)I_n$.
References:
[1] Aouchiche, M., Hansen, P.: Two Laplacians for the distance matrix of a graph. Linear Algebra Appl. 439 (2013), 21-33. DOI 10.1016/j.laa.2013.02.030 | MR 3045220 | Zbl 1282.05086
[2] Aouchiche, M., Hansen, P.: Distance spectra of graphs: A survey. Linear Algebra Appl. 458 (2014), 301-386. DOI 10.1016/j.laa.2014.06.010 | MR 3231823 | Zbl 1295.05093
[3] Aouchiche, M., Hansen, P.: Some properties of the distance Laplacian eigenvalues of a graph. Czech. Math. J. 64 (2014), 751-761. DOI 10.1007/s10587-014-0129-2 | MR 3298557 | Zbl 1349.05083
[4] Brouwer, A. E., Haemers, W. H.: Spectra of graphs. Universitext. Berlin: Springer (2012). DOI 10.1007/978-1-4614-1939-6 | MR 2882891 | Zbl 1231.05001
[5] Cvetković, D. M., Doob, M., Sachs, H.: Spectra of Graphs: Theory and Application. Pure and Applied Mathematics 87. Academic Press, New York (1980). MR 0572262 | Zbl 0824.05046
[6] Das, K. C., Aouchiche, M., Hansen, P.: On (distance) Laplacian energy and (distance) signless Laplacian energy of graphs. Discrete Appl. Math. 243 (2018), 172-185. DOI 10.1016/j.dam.2018.01.004 | MR 3804748 | Zbl 1387.05147
[7] Díaz, R. C., Rojo, O.: Sharp upper bounds on the distance energies of a graph. Linear Algebra Appl. 545 (2018), 55-75. DOI 10.1016/j.laa.2018.01.032 | MR 3769113 | Zbl 1390.05124
[8] Ganie, H. A.: On distance Laplacian spectrum (energy) of graphs. Discrete Math. Algorithms Appl. 12 (2020), Article ID 2050061, 16 pages. DOI 10.1142/S1793830920500615 | MR 4157019 | Zbl 1457.05064
[9] Ganie, H. A.: On the distance Laplacian energy ordering of tree. Appl. Math. Comput. 394 (2021), Article ID 125762, 10 pages. DOI 10.1016/j.amc.2020.125762 | MR 4182919 | Zbl 1462.05222
[10] Ganie, H. A., Chat, B. A., Pirzada, S.: Signless Laplacian energy of a graph and energy of line graph. Linear Algebra Appl. 544 (2018), 306-324. DOI 10.1016/j.laa.2018.01.021 | MR 3765789 | Zbl 1388.05114
[11] Ganie, H. A., Pirzada, S., Rather, B. A., Trevisan, V.: Further developments on Brouwer's conjecture for the sum of Laplacian eigenvalues of graphs. Linear Algebra Appl. 588 (2020), 1-18. DOI 10.1016/j.laa.2019.11.020 | MR 4037607 | Zbl 1437.05139
[12] Gutman, I., Zhou, B.: Laplacian energy of a graph. Linear Algebra Appl. 414 (2006), 29-37. DOI 10.1016/j.laa.2005.09.008 | MR 2209232 | Zbl 1092.05045
[13] Indulal, G., Gutman, I., Vijayakumar, A.: On distance energy of graphs. MATCH Commun. Math. Comput. Chem. 60 (2008), 461-472. MR 2457864 | Zbl 1199.05226
[14] Li, X., Shi, Y., Gutman, I.: Graph Energy. Springer, New York (2012). DOI 10.1007/978-1-4614-4220-2 | MR 2953171 | Zbl 1262.05100
[15] Monsalve, J., Rada, J.: Oriented bipartite graphs with minimal trace norm. Linear Multilinear Algebra 67 (2019), 1121-1131. DOI 10.1080/03081087.2018.1448051 | MR 3937031 | Zbl 1411.05172
[16] Pirzada, S.: An Introduction to Graph Theory. Orient Blackswan, Hyderabad (2012).
[17] Pirzada, S., Ganie, H. A.: On the Laplacian eigenvalues of a graph and Laplacian energy. Linear Algebra Appl. 486 (2015), 454-468. DOI 10.1016/j.laa.2015.08.032 | MR 3401774 | Zbl 1327.05157
[18] Yang, J., You, L., Gutman, I.: Bounds on the distance Laplacian energy of graphs. Kragujevac J. Math. 37 (2013), 245-255. MR 3150862 | Zbl 1299.05236
Search
Partner of
