Distance signless Laplacian eigenvalues of graphs

doi:10.1007/s11464-019-0779-3

Front. Math. China

2019, Vol. 14

Issue (4) : 693-713 https://doi.org/10.1007/s11464-019-0779-3

RESEARCH ARTICLE

Distance signless Laplacian eigenvalues of graphs

Kinkar Chandra DAS¹(

), Huiqiu LIN², Jiming GUO²

¹. Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
². Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

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Abstract

Suppose that the vertex set of a graph G is $V (G) = {v 1, v 2, ..., v n}$ . The transmission $T r (v i)$ (or D_i) of vertex vi is defined to be the sum of distances from vi to all other vertices. Let $T r (G)$ be the $n × n$ diagonal matrix with its (i, i)-entry equal to $T r G (v i)$ . The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as $L (G) = T r (G) + D (G)$ , where $D (G)$ is the distance matrix of G. In this paper, we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible. We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs. Moreover, we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees, and characterize extremal graphs.

Keywords Graph distance signless Laplacian spectral radius second largest eigenvalue of distance signless Laplacian matrix spread

Corresponding Author(s): Kinkar Chandra DAS

Issue Date: 23 September 2019

Cite this article:

Kinkar Chandra DAS,Huiqiu LIN,Jiming GUO. Distance signless Laplacian eigenvalues of graphs[J]. Front. Math. China, 2019, 14(4): 693-713.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-019-0779-3
https://academic.hep.com.cn/fmc/EN/Y2019/V14/I4/693

1	M Aouchiche, P Hansen. A signless Laplacian for the distance matrix of a graph. Preprint
2	D Cvetković, M Doob, H Sachs. Spectra of Graphs-Theory and Applications. 3rd ed. Heidelberg: Johann Ambrosius Barth Verlag, 1995
3	K C Das. Proof of conjectures on the distance signless Laplacian eigenvalues of graphs. Linear Algebra Appl, 2015, 467: 100–115 https://doi.org/10.1016/j.laa.2014.11.008
4	H Dong, X Guo. Ordering trees by their Wiener indices. MATCH Commun Math Comput Chem, 2006, 56: 527–540
5	WH Haemers. Interlacing eigenvalues and graphs. Linear Algebra Appl, 1995, 226-228: 593–616 https://doi.org/10.1016/0024-3795(95)00199-2
6	W Hong, L You. Some sharp bounds on the distance signless Laplacian spectral radius of graphs. Preprint
7	G Indulal. Sharp bounds on the distance spectral radius and the distance energy of graphs. Linear Algebra Appl, 2009, 430: 106–113 https://doi.org/10.1016/j.laa.2008.07.005
8	H Lin, K C Das. Characterization of extremal graphs from distance signless Laplacian eigenvalues. Linear Algebra Appl, 2016, 500: 77–87 https://doi.org/10.1016/j.laa.2016.03.017
9	H Lin, X Lu. Bounds on the distance signless Laplacian spectral radius in terms of clique number. Linear Multilinear Algebra, 2015, 63(9): 1750–1759 https://doi.org/10.1080/03081087.2014.972393
10	F Tian, X Li, J Rou. A note on the signless Laplacian and distance signless Laplacian eigenvalues of graphs. J Math Res Appl, 2014, 34(6): 647–654
11	Wolfram Research Inc, Mathematica. version 7.0. Champaign, IL, 2008
12	R Xing, B Zhou. On the distance and distance signless Laplacian spectral radii of bicyclic graphs. Linear Algebra Appl, 2013, 439: 3955–3963 https://doi.org/10.1016/j.laa.2013.10.005
13	R Xing, B Zhou, J Li. On the distance signless Laplacian spectral radius of graphs. Linear Multilinear Algebra, 2014, 62: 1377{1387 https://doi.org/10.1080/03081087.2013.828720
14	F Zhang. Matrix Theory: Basic Results and Techniques. New York: Springer-Verlag, 1999 https://doi.org/10.1007/978-1-4757-5797-2

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