On the normalized Laplacian spectral radii of a graph and its line graph

Shaowei Sun; Kinkar Chandra Das

doi:10.1007/s40314-020-01340-2

On the normalized Laplacian spectral radii of a graph and its line graph

Shaowei Sun, Kinkar Chandra Das

Department of Mathematics

Zhejiang University of Science and Technology

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Normalized Laplacian eigenvalues are very popular in spectral graph theory. The normalized Laplacian spectral radius ρ₁(G) of a graph G is the largest eigenvalue of normalized Laplacian matrix of G. In this paper, we determine the extremal graph for the minimum normalized Laplacian spectral radii of nearly complete graphs. We present several lower bounds on ρ₁(G) in terms of graph parameters and characterize the extremal graphs. Still, there is no result on the normalized Laplacian eigenvalues of line graphs. Here, we obtain sharp lower bounds on the normalized Laplacian spectral radii of line graphs. Moreover, we compare ρ₁(G) and ρ₁(L_G) (LGisthelinegraphofG) in some class of graphs as they are incomparable in the general case. Finally, we present a relation on the normalized Laplacian spectral radii of a graph and its line graph.

Original language	English
Article number	283
Journal	Computational and Applied Mathematics
Volume	39
Issue number	4
DOIs	https://doi.org/10.1007/s40314-020-01340-2
State	Published - 1 Dec 2020

Keywords

Bipartite edge frustration
Independence number
Line graph
Nearly complete graph
Normalized Laplacian spectral radius

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10.1007/s40314-020-01340-2

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title = "On the normalized Laplacian spectral radii of a graph and its line graph",

abstract = "Normalized Laplacian eigenvalues are very popular in spectral graph theory. The normalized Laplacian spectral radius ρ1(G) of a graph G is the largest eigenvalue of normalized Laplacian matrix of G. In this paper, we determine the extremal graph for the minimum normalized Laplacian spectral radii of nearly complete graphs. We present several lower bounds on ρ1(G) in terms of graph parameters and characterize the extremal graphs. Still, there is no result on the normalized Laplacian eigenvalues of line graphs. Here, we obtain sharp lower bounds on the normalized Laplacian spectral radii of line graphs. Moreover, we compare ρ1(G) and ρ1(LG) (LGisthelinegraphofG) in some class of graphs as they are incomparable in the general case. Finally, we present a relation on the normalized Laplacian spectral radii of a graph and its line graph.",

keywords = "Bipartite edge frustration, Independence number, Line graph, Nearly complete graph, Normalized Laplacian spectral radius",

author = "Shaowei Sun and Das, \{Kinkar Chandra\}",

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doi = "10.1007/s40314-020-01340-2",

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T1 - On the normalized Laplacian spectral radii of a graph and its line graph

AU - Sun, Shaowei

AU - Das, Kinkar Chandra

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Y1 - 2020/12/1

N2 - Normalized Laplacian eigenvalues are very popular in spectral graph theory. The normalized Laplacian spectral radius ρ1(G) of a graph G is the largest eigenvalue of normalized Laplacian matrix of G. In this paper, we determine the extremal graph for the minimum normalized Laplacian spectral radii of nearly complete graphs. We present several lower bounds on ρ1(G) in terms of graph parameters and characterize the extremal graphs. Still, there is no result on the normalized Laplacian eigenvalues of line graphs. Here, we obtain sharp lower bounds on the normalized Laplacian spectral radii of line graphs. Moreover, we compare ρ1(G) and ρ1(LG) (LGisthelinegraphofG) in some class of graphs as they are incomparable in the general case. Finally, we present a relation on the normalized Laplacian spectral radii of a graph and its line graph.

AB - Normalized Laplacian eigenvalues are very popular in spectral graph theory. The normalized Laplacian spectral radius ρ1(G) of a graph G is the largest eigenvalue of normalized Laplacian matrix of G. In this paper, we determine the extremal graph for the minimum normalized Laplacian spectral radii of nearly complete graphs. We present several lower bounds on ρ1(G) in terms of graph parameters and characterize the extremal graphs. Still, there is no result on the normalized Laplacian eigenvalues of line graphs. Here, we obtain sharp lower bounds on the normalized Laplacian spectral radii of line graphs. Moreover, we compare ρ1(G) and ρ1(LG) (LGisthelinegraphofG) in some class of graphs as they are incomparable in the general case. Finally, we present a relation on the normalized Laplacian spectral radii of a graph and its line graph.

KW - Bipartite edge frustration

KW - Independence number

KW - Line graph

KW - Nearly complete graph

KW - Normalized Laplacian spectral radius

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DO - 10.1007/s40314-020-01340-2

M3 - Article

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VL - 39

JO - Computational and Applied Mathematics

JF - Computational and Applied Mathematics

IS - 4

M1 - 283

ER -

On the normalized Laplacian spectral radii of a graph and its line graph

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Keywords

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