On trees with algebraic connectivity greater than or equal to $$2\left( 1-\cos \left( \frac{\pi }{7}\right) \right) $$ 2 1 - cos π 7

Let T be a tree on n vertices and algebraic connectivity $$\alpha (T).$$ The trees on $$n\ge 45$$ vertices and $$\alpha (T)\ge \frac{5-\sqrt{21}}{2}$$ have already been completely characterized. In case of $$\alpha (T)\ge 2-\sqrt{3}$$ , it was proved that this set of trees can be partitioned in six classes, $$C_i,$$ $$1 \le i \le 6,$$ ordered by algebraic connectivity, in the sense that $$\alpha (T_i)>\alpha (T_j)$$ whenever $$T_i \in C_i,$$ $$T_j \in C_{j},$$ and $$1 \le i更多