We observed the effects on the spectral radius of submatrices of the Laplacian matrix L for a tree by deleting a row and column of L corresponding to a vertex of the tree. This enabled us to classify trees as either of Type A or Type B. In this paper, we extend these results to graphs which are not trees and offer a similar classification. Additionally, we show counterexamples to theorems that are true for trees, but not so for general graphs.