Six classes of trees with largest normalized algebraic connectivity

The normalized algebraic connectivity of a graph G, denoted by λ2(G), is the second smallest eigenvalue of its normalized Laplacian matrix. In this paper, we firstly determine all trees with λ2(G)⩾1−63. Then we classify such trees into six classes C1,…,C6 and prove that λ2(Ti)>λ2(Tj) for 1⩽i