Principal eigenvectors of irregular graphs

Main Article Content

Sebastian M. Cioabă
David A. Gregory

Abstract

Let G be a connected graph. Thispaper studiesthe extreme entriesof the principal
eigenvector x of G, the unique positive unit eigenvector corresponding to the greatest eigenvalue λ1
of the adjacency matrix of G. If G hasmaximum degree ∆, the greatest entry xmax of x is at most
1/√1 + λ21/∆. This improves a result of Papendieck and Recht. The least entry xmin of x as well
asthe principal ratio xmax/xmin are studied. It is conjectured that for connected graphs of order
n ≥ 3, the principal ratio isalwaysattained by one of the lollipop graphs obtained by attaching a
path graph to a vertex of a complete graph

Article Details

Section
Article