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 yie

There is a contrast between the two sets of functional equationsimageandimagesatisfied by the even and odd components of a solution of f(x + y) = f(x)f(y). Schwaiger and, later, Förg-Rob and Schwaiger considered the extension of these ideas to the case where f is sum of n components. Here we shorten and simplify the statements and proofs of some of these results by a more systematic use of matrix notation.