The Spectral Radius of Submatrices of Laplacian Matrices for Graphs with Cut Vertices
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.
Molitierno, J.J. (2008). The spectral radius of submatrices of laplacian matrices for graphs with cut vertices. Linear Algebra & Its Applications, 428(8/9), 1987-1999. doi: 10.1016/j.laa.2007.11.008