ON SPECTRA OF POWER GRAPHS OF FINITE CYCLIC AND DIHEDRAL GROUPS

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,v∈𝒫(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 ℤn and the dihedral group Dn. We provide upper and lower bounds on the largest eigenvalue of the distance signless Laplacian matrix of 𝒫(ℤn) and 𝒫(Dn). We also give a short proof of the lower bound on the algebraic connectivity of 𝒫(ℤn) obtained by Chattopadhyay and Panigrahi (Linear and Multilinear Algebra 63:7 (2015), 1345–1355).