Sandbox talk:DavidKohler/Relativized Alon Conjecture/Regular tree completion

I have notes saying that using the correspondence I can simplify the proof quite a bit. In addition, it would be interesting to add an argument for why

$\displaystyle \lambda _{1}({\text{Tree}}_{d}(\psi ))\geq 2{\sqrt {d-1}}$

for any (connected) graph ψ.