On Trees With Algebraic Connectivity Greater Than Or Equal To 2(1-Cos(Pi/7))

Let T be a tree on n vertices and algebraic connectivity alpha(T). The trees on n >= 45 vertices and a( T) >= 5-root 21/2 have already been completely characterized. In case of alpha( T) >= 2- root 3, it was proved that this set of trees can be partitioned in six classes, C-i, 1 <= i <= 6, ordered by algebraic connectivity, in the sense that alpha( T-i) > alpha(T-j) whenever T-i is an element of C-i, T-j is an element of C-j, and 1 <= i < j <= 6. In this work, we extend both results by studying four classes of trees, C-i, 7 <= i <= 10. We prove that, for n >= 45, 2 (1 - cos (pi/7)) <= alpha(T) < 5-root 21/2 if and only if T is an element of C-9 boolean OR C-10. Moreover, the set of trees with 2 (1 - cos (pi/7)) <= alpha(T) < 2 - root 3 can be partitioned in the classes C-i, 7 <= i <= 10, also ordered by algebraic connectivity.