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

Abstract

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 $1 \le i<j \le 6.$ In this work, we extend both results by studying four classes of trees, $C_i,$ $7 \le i \le 10.$ We prove that, for $n\ge 45,$ $2\left( 1-\cos \left( \frac{\pi }{7}\right) \right) \le \alpha (T)<\frac{5-\sqrt{21}}{2}$ if and only if $T \in C_9 \cup C_{10}$. Moreover, the set of trees with $2\left( 1-\cos \left( \frac{\pi }{7}\right) \right) \le \alpha (T)< 2 - \sqrt{3}$ can be partitioned in the classes $C_i$, $7 \le i \le 10$, also ordered by algebraic connectivity.