The six classes of trees with the largest algebraic connectivity

In this paper, we study the algebraic connectivity α(T)α(T) of a tree T  . We introduce six Classes (C1)(C1)–(C6)(C6) of trees of order n, and prove that if T   is a tree of order n⩾15n⩾15, then α(T)⩾2-3 if and only if T∈⋃i=16Ci, where the equality holds if and only if T   is a tree in the Class (C6)(C6). At the same time we give a complete ordering of the trees in these six classes by their algebraic connectivity. In particular, we show that α(Ti)>α(Tj)α(Ti)>α(Tj) if 1⩽i