Algebraic Connectivity for Subclasses of Caterpillars

Ordering of subclasses of trees by algebraic connectivity is a very active area of research. Let G = (V,E) be a simple undirected graph on n vertices. The Laplacian matrix of G is the n × n matrix L (G) = D (G) − A (G) where A (G) is the adjacency matrix and D (G) is the diagonal matrix of vertex degrees. It is well known that L (G) is a positive semidefinite matrix and that (0, e) is an eigenpair of L (G) where e is the all ones vector. Let us denote the eigenvalues of L (G) by 0 = λn (G) ≤ λn−1 (G) ≤ · · · ≤ λ2 (G) ≤ λ1 (G) .