= Solution
There is one constant path and $r$ loop generators; every other path is a word in those loops. Distinct words are distinct basis paths, so there are no relations among the generators. Therefore
$$
\boxed{kL_r\cong k\langle x_1,\ldots,x_r\rangle,}
$$
the <free associative algebra> on $r$ generators, with each loop sent to its corresponding generator. For $r=1$ this is $k[x_1]$; for $r\geq2$ it is noncommutative. The <quiver with one loop> is the $r=1$ case of the <quiver with r loops>.
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