  In this paper we provide a necessary and sufficient condition for the
amenability of the automorphism group of Fra\"iss\'e structures and apply it to
prove the non-amenability of the automorphism groups of the directed graph
$\mathbf{S}(3)$ and the Boron tree structure $\mathbf{T}$. Also, we provide a
negative answer to the Unique Ergodicity-Generic Point problem of
Angel-Kechris-Lyons [AKL]. By considering $\mathrm{GL}(\mathbf{V}_\infty)$,
where $\mathbf{V}_\infty$ is the countably infinite dimensional vector space
over a finite field $F_q$, we show that the unique invariant measure on the
universal minimal flow of $\mathrm{GL}(\mathbf{V}_\infty)$ is not supported on
the generic orbit.
