Abstract
Let G be a connected graph. Thispaper studiesthe extreme entriesof the principaleigenvector x of G, the unique positive unit eigenvector corresponding to the greatest eigenvalue λ1of the adjacency matrix of G. If G hasmaximum degree ∆, the greatest entry xmax of x is at most1/√1 + λ21/∆. This improves a result of Papendieck and Recht. The least entry xmin of x as wellasthe principal ratio xmax/xmin are studied. It is conjectured that for connected graphs of ordern ≥ 3, the principal ratio isalwaysattained by one of the lollipop graphs obtained by attaching apath graph to a vertex of a complete graph
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