Summary: 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 *1 of the adjacency matrix of G. If G has maximum degree $\Delta $, the greatest entry xmax of x is at most 1/q1 + *$21/\Delta $. 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.

05C50, 15A18

spectral radius, irregular graph, eigenvectors