Principal eigenvectors of irregular graphs

Abstract

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 λ 1 of the adjacency matrix of G. If G has maximum degree ∆, the greatest entry xmax of x is at most 1/ Õ 1 + λ 2 1 /∆. This improves a result of Papendieck and Recht. The least entry x min of x as well as the principal ratio xmax/x min 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.

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