On the eigenvalues of Aα-matrix of graphs

Shuting Liu; Kinkar Chandra Das; Jinlong Shu

doi:10.1016/j.disc.2020.111917

On the eigenvalues of A_α-matrix of graphs

Shuting Liu
, Kinkar Chandra Das
, Jinlong Shu

Department of Mathematics

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

20 Scopus citations

Abstract

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α∈[0,1], Nikiforov defined the matrix A_α(G) as A_α(G)=αD(G)+(1−α)A(G). In this paper, we study the kth largest eigenvalue of A_α-matrix of graphs, where 1≤k≤n. We present several upper and lower bounds on the kth largest eigenvalue of A_α-matrix and characterize the extremal graphs corresponding to some of these obtained bounds. As applications, some bounds we obtained can generalize some known results on adjacency matrix and signless Laplacian matrix of graphs. Finally, we solve a problem proposed by Nikiforov (2017).

Original language	English
Article number	111917
Journal	Discrete Mathematics
Volume	343
Issue number	8
DOIs	https://doi.org/10.1016/j.disc.2020.111917
State	Published - Aug 2020

Keywords

A-spectral radius
Degree
Graph
The kth largest eigenvalue of A(G)

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10.1016/j.disc.2020.111917

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title = "On the eigenvalues of Aα-matrix of graphs",

abstract = "Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α∈[0,1], Nikiforov defined the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G). In this paper, we study the kth largest eigenvalue of Aα-matrix of graphs, where 1≤k≤n. We present several upper and lower bounds on the kth largest eigenvalue of Aα-matrix and characterize the extremal graphs corresponding to some of these obtained bounds. As applications, some bounds we obtained can generalize some known results on adjacency matrix and signless Laplacian matrix of graphs. Finally, we solve a problem proposed by Nikiforov (2017).",

keywords = "A-spectral radius, Degree, Graph, The kth largest eigenvalue of A(G)",

author = "Shuting Liu and Das, \{Kinkar Chandra\} and Jinlong Shu",

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year = "2020",

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doi = "10.1016/j.disc.2020.111917",

language = "English",

volume = "343",

journal = "Discrete Mathematics",

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T1 - On the eigenvalues of Aα-matrix of graphs

AU - Liu, Shuting

AU - Das, Kinkar Chandra

AU - Shu, Jinlong

PY - 2020/8

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N2 - Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α∈[0,1], Nikiforov defined the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G). In this paper, we study the kth largest eigenvalue of Aα-matrix of graphs, where 1≤k≤n. We present several upper and lower bounds on the kth largest eigenvalue of Aα-matrix and characterize the extremal graphs corresponding to some of these obtained bounds. As applications, some bounds we obtained can generalize some known results on adjacency matrix and signless Laplacian matrix of graphs. Finally, we solve a problem proposed by Nikiforov (2017).

AB - Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α∈[0,1], Nikiforov defined the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G). In this paper, we study the kth largest eigenvalue of Aα-matrix of graphs, where 1≤k≤n. We present several upper and lower bounds on the kth largest eigenvalue of Aα-matrix and characterize the extremal graphs corresponding to some of these obtained bounds. As applications, some bounds we obtained can generalize some known results on adjacency matrix and signless Laplacian matrix of graphs. Finally, we solve a problem proposed by Nikiforov (2017).

KW - A-spectral radius

KW - Degree

KW - Graph

KW - The kth largest eigenvalue of A(G)

UR - https://www.scopus.com/pages/publications/85082703530

U2 - 10.1016/j.disc.2020.111917

DO - 10.1016/j.disc.2020.111917

M3 - Article

AN - SCOPUS:85082703530

SN - 0012-365X

VL - 343

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 8

M1 - 111917

ER -

On the eigenvalues of A_α-matrix of graphs

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Keywords

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