April 2023 ON SPECTRA OF POWER GRAPHS OF FINITE CYCLIC AND DIHEDRAL GROUPS
Subarsha Banerjee, Avishek Adhikari
Rocky Mountain J. Math. 53(2): 341-356 (April 2023). DOI: 10.1216/rmj.2023.53.341

Abstract

The power graph 𝒫(G) of a finite group G is defined to be the graph whose vertex set is G and in which two distinct vertices u,v𝒫(G) are adjacent if and only if u=vm or v=un for some positive integers m,n. The distance signless Laplacian matrix of a graph 𝒢, denoted by DQ(𝒢), is defined as DQ(𝒢)=Tr(𝒢)+D(𝒢), where D(𝒢) is the distance matrix of 𝒢 and Tr(𝒢) is the transmission matrix of 𝒢. We determine the distance signless Laplacian eigenvalues of the power graphs of the finite cyclic group n and the dihedral group Dn. We provide upper and lower bounds on the largest eigenvalue of the distance signless Laplacian matrix of 𝒫(n) and 𝒫(Dn). We also give a short proof of the lower bound on the algebraic connectivity of 𝒫(n) obtained by Chattopadhyay and Panigrahi (Linear and Multilinear Algebra 63:7 (2015), 1345–1355).

Citation

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Subarsha Banerjee. Avishek Adhikari. "ON SPECTRA OF POWER GRAPHS OF FINITE CYCLIC AND DIHEDRAL GROUPS." Rocky Mountain J. Math. 53 (2) 341 - 356, April 2023. https://doi.org/10.1216/rmj.2023.53.341

Information

Received: 1 May 2020; Revised: 21 January 2022; Accepted: 31 May 2022; Published: April 2023
First available in Project Euclid: 20 June 2023

MathSciNet: MR4604760
zbMATH: 1519.05146
Digital Object Identifier: 10.1216/rmj.2023.53.341

Subjects:
Primary: 05C25
Secondary: 05C50

Keywords: algebraic connectivity , distance signless Laplacian , finite groups , graph spectra , power graph , spectral radius

Rights: Copyright © 2023 Rocky Mountain Mathematics Consortium

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Vol.53 • No. 2 • April 2023
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