ELA, Volume 16, pp. 366-379, October 2007, abstract.

Principal Eigenvectors of Irregular Graphs

Sebastian M. Cioaba and David A. Gregory

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 lambda_1 of the adjacency matrix 
of G. If G has maximum degree Delta, the greatest entry x_max of x is 
at most 1/sqrt(1+(lambda_1)^2/Delta). This improves a result of Papendieck 
and Recht. The least entry x_min of x as well as the principal ratio 
x_max/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.