Multiplicities of distance Laplacian eigenvalues and forbidden subgraphs

Fernandes, Rosário; Freitas, Maria Aguieiras A. de; Silva, Celso M. da; Del-Vecchio, Renata R.

doi:10.1016/j.laa.2017.11.031

Cited by 11 publications

(3 citation statements)

References 9 publications

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“…Fernandes et al [6] and Lu et al [7] determined the graphs having distance Laplacian spectral radius of multiplicity n − 3. Further, the investigation of the graphs having some distance Laplacian eigenvalue of multiplicity n − 3 was done by Ma et al in [8].…”

Section: Aouchiche and Hansenmentioning

confidence: 99%

On graphs with distance Laplacian eigenvalues of multiplicity $n-4$

Khan¹,

Pirzada²

2022

Preprint

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Let G be a connected simple graph with n vertices. The distance Laplacian matrix D L (G) is defined as D L (G) = Diag(T r) − D(G), where Diag(T r) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix of G. The eigenvalues of D L (G) are the distance Laplacian eigenvalues of G and are denoted byis called the distance Laplacian spectral radius. Lu et al. (2017), Fernandes et al. (2018) and Ma et al. (2018) completely characterized the graphs having some distance Laplacian eigenvalue of multiplicity n − 3. In this paper, we characterize the graphs having distance Laplacian spectral radius of multiplicity n − 4 together with one of the distance Laplacian eigenvalue as n of multiplicity either 3 or 2. Further, we completely determine the graphs for which the distance Laplacian eigenvalue n is of multiplicity n − 4.

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Section: Aouchiche and Hansenmentioning

confidence: 99%

On graphs with distance Laplacian eigenvalues of multiplicity $n-4$

Khan¹,

Pirzada²

2022

Preprint

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“…The normalized Laplacian spectrum of graphs has been studied intensively (see [1][2][3][4][5][6][7][8]), because it reveals some structural properties and some relevant dynamical aspects (such as random walk) of graphs [9]. Recently, graphs with some eigenvalue of large multiplicity have attracted much attention (see [10][11][12]). However, there are few results on the normalized Laplacian eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

On graphs with some normalized Laplacian eigenvalue of extremal multiplicity

Tian,

Cai,

Liang

et al. 2020

Preprint

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Let G be a connected simple graph on n vertices. Let L(G) be the normalized Laplacian matrix of G and ρ n−1 (G) be the second least eigenvalue of L(G). Denote by ν(G) the independence number of G. Recently, the paper [Characterization of graphs with some normalized Laplacian eigenvalue of multiplicity n − 3, arXiv:1912.13227] discussed the graphs with some normalized Laplacian eigenvalue of multiplicity n − 3. However, there is one remaining case (graphs with ρ n−1 (G) = 1 and ν(G) = 2) not considered. In this paper, we focus on cographs and graphs with diameter 3 to investigate the graphs with some normalized Laplacian eigenvalue of multiplicity n − 3.

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“…In [26], chromatic number is used to derive a lower bound for distance signless Laplacian spectral radius. The distance signless Laplacian spectrum has varies connections with other interesting graph topics such as chromatic number [10]; domination and independence numbers [21], Estrada indices [4,5,22,23,[34][35][36]38], cospectrality [11,42], multiplicity of the distance (signless) Laplacian eigenvalues [25,29,30] and many more, see e.g., [1][2][3]27,28,32].…”

Section: Introductionmentioning

confidence: 99%

Bounds for the Generalized Distance Eigenvalues of a Graph

2019

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Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and T r ( G ) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by D α ( G ) the generalized distance matrix, i.e., D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue.

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Multiplicities of distance Laplacian eigenvalues and forbidden subgraphs

Cited by 11 publications

References 9 publications

On graphs with distance Laplacian eigenvalues of multiplicity $n-4$

On graphs with distance Laplacian eigenvalues of multiplicity $n-4$

On graphs with some normalized Laplacian eigenvalue of extremal multiplicity

Bounds for the Generalized Distance Eigenvalues of a Graph

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