Sandbox:DavidKohler/Alon try 1/Theorem 3.13

This is the theorem described in section 3.8 as a curious theorem. It plays a role in describing tangles.

Statement

Theorem 3.13
Let $d\geq 3$ and let $G$ be a finite connected graph whose maximal degree is less or equal to $d$. Then the following holde: $\lambda _{1}({\textrm {Tree}}_{d}(G))=2{\sqrt {d-1}}\iff \lambda _{\textrm {Irred}}(G)\leq {\sqrt {d-1}}$ $\lambda _{1}({\textrm {Tree}}_{d}(G))>2{\sqrt {d-1}}\iff \lambda _{\textrm {Irred}}(G)>{\sqrt {d-1}}$