Need some help with Gabriel's theorem, doing the part "if a graph is of finite type, it must be a Dynkin graph" like this:
Let $\vec{Q}$ be of finite type. All representations is a direct sum of indecomposables representations.
Let $v\in \mathbb{Z}_+\times \cdots \times \mathbb{Z}_+=\mathbb{Z}_+^I$ where $I$ is the number of vertices of the quiver.
Exists a finite number of isomorphism classes os representations with dimension $v$. If $V_1,\cdots, V_s$ are the indecomposables representations, we have $$V_1^{d_1}\oplus \cdots \oplus V_s^{d_s}.$$
We have only a finite number of choices $d_1,\cdots , d_s$ to have a representation with dimension $v$.
Then I have this property: two elements $x, x' \in R(v)$ define isomorphic representations of $\vec{Q}$ iff they are in the same $GL(v)$ orbit.
That give me this: the action of $GL(v)$ on $R(\vec{Q}, v)$ has only a finite number of orbits.
Then I have that exists a orbit $O_x$ with dim$O_x=$dim$R(v)$, so $$\text{dim}GL(v)-\text{dim}O_x=\text{dim}G_x$$ then $$\sum_{i\in I}v_i^2-\text{dim}R(v)=\text{dim}G_x$$ and $$G_x=\{ g\in GL(v);g(x)=x \}.$$
Now we have that, $$\sum_{i\in I}v_i^2-\sum_h v_{s(h)} v_{t(h)}\geq 1$$ $$\sum_{i\in I}v_i^2-\sum_h v_{s(h)} v_{t(h)}=<v,v>=q(v)\geq 1$$ then $q$ is positive defined, so $\vec{Q}$ is dinkyn.
This all is true for $v\in \mathbb{Z}_+\times \cdots \times \mathbb{Z}_+=\mathbb{Z}_+^I$, how do I show that it's true for $v\in \mathbb{Z}\times \cdots \times \mathbb{Z}=\mathbb{Z}^I$?