The spectral radius of submatrices of Laplacian matrices for trees and its comparison to the fiedler vector

We consider the effects on the spectral radius of submatrices of the Laplacian matrix for graphs by deleting the row and column corresponding to various vertices of the graph. We focus most of our attention on trees and determine which vertices v will yield the maximum and minimum spectral radius of the Laplacian when row v and column v are deleted. At this point, comparisons are made between these results and results concerning the Fiedler vector of the tree.