Principal eigenvectors of irregular graphs

Abstract

Let G be a connected graph. This paper studies the extreme entries of 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 has maximum degree A, the greatest entry xmax of x is at most 1/ √1+λ12/Δ. This improves a result of Papendieck and Recht. The least entry xmin of x as well as the principal ratio xmax/xmin are studied. It is conjectured that for connected graphs of order n ≥ 3, the principal ratio is always attained by one of the lollipop graphs obtained by attaching a path graph to a vertex of a complete graph.