Proof for four conjectures about the distance Laplacian and distance signless Laplacian eigenvalues of a graph

Tian, Fenglei; Wong, Dein; Rou, Jianling

doi:10.1016/j.laa.2014.12.015

Cited by 23 publications

(12 citation statements)

References 9 publications

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“…Lin and Zhou [8] determined some extremal graphs with minimum distance Laplacian spectral radius among several classes of graphs: graphs with fixed number of pendent vertices, trees with fixed bipartition, graphs with fixed edge connectivity at most half of the number of vertices, graphs with fixed connectivity. Tian et al [11] proved four conjectures proposed by Aouchiche and Hansen in [3]. For the articles on the distance signless Laplacian eigenvalues of graphs, one can see [2,6,7,13,14] for details.…”

Section: Introductionmentioning

confidence: 96%

Lower bounds of distance Laplacian spectral radii of n-vertex graphs in terms of matching number

Tian

Wong

2016

Linear Algebra and its Applications

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Section: Introductionmentioning

confidence: 96%

Lower bounds of distance Laplacian spectral radii of n-vertex graphs in terms of matching number

Tian

Wong

2016

Linear Algebra and its Applications

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“…Finally, the generalized distance spectra of some graphs obtained by operations are also studied. For more review about distance Laplacian spectrum and distance signless Laplacian spectrum of graphs, readers may refer to [12][13][14][15][16] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Distance matrix correlation spectrum of graphs

Lu¹,

Liu²

2020

Preprint

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Let G be a simple, connected graph, D(G) be the distance matrix of G, and T r(G) be the diagonal matrix of vertex transmissions of G. The distance Laplacian matrix and distance signless Laplacian matrix of G are defined byand the generalized distance spectral radius of G is the largest eigenvalue of D α (G). In this paper, we give a complete description of the D−spectrum, L−spectrum and Q−spectrum of some graphs obtained by operations. In addition, we present some new upper and lower bounds on the generalized distance spectral radius of G and of its line graph L(G), based on other graph-theoretic parameters, and characterize the extremal graphs. Finally, we study the generalized distance spectrum of some composite graphs.

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“…Li et al [8] gave a lower bound on the distance signless Laplacian spectral radius in terms of chromatic number and determined the unique graph with minimum distance signless Laplacian spectral radius among connected graphs with fixed number of vertices and chromatic number. For more review about distance (signless) Laplacian spectral radius of graphs, readers may refer to [1,3,6,9,14,15] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On the generalized distance spectral radius of graphs

Cui,

Tian,

Zheng

2019

Preprint

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The generalized distance spectral radius of a connected graph G is the spectral radius of the generalized distance matrix of G, defined bywhere D(G) and T r(G) denote the distance matrix and diagonal matrix of the vertex transmissions of G, respectively. This paper characterizes the unique graph with minimum generalized distance spectral radius among the connected graphs with fixed chromatic number, which answers a question about the generalized distance spectral radius in spectral extremal theories. In addition, we also determine graphs with minimum generalized distance spectral radius among the n-vertex trees and unicyclic graphs, respectively. These results generalize some known results about distance spectral radius and distance signless Laplacian spectral radius of graphs.

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Proof for four conjectures about the distance Laplacian and distance signless Laplacian eigenvalues of a graph

Cited by 23 publications

References 9 publications

Lower bounds of distance Laplacian spectral radii of n-vertex graphs in terms of matching number

Lower bounds of distance Laplacian spectral radii of n-vertex graphs in terms of matching number

Distance matrix correlation spectrum of graphs

On the generalized distance spectral radius of graphs

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