  The Fomin-Kirillov algebra $\mathcal E_n$ is a noncommutative quadratic
algebra with a generator for every edge of the complete graph on $n$ vertices.
For any graph $G$ on $n$ vertices, we define $\mathcal E_G$ to be the
subalgebra of $\mathcal E_n$ generated by the edges of $G$. We show that these
algebras have many parallels with Coxeter groups and their nil-Coxeter
algebras: for instance, $\mathcal E_G$ is a free $\mathcal E_H$-module for any
$H\subseteq G$, and if $\mathcal E_G$ is finite-dimensional, then its Hilbert
series has symmetric coefficients. We determine explicit monomial bases and
Hilbert series for $\mathcal E_G$ when $G$ is a simply-laced finite Dynkin
diagram or a cycle, in particular showing that $\mathcal E_G$ is
finite-dimensional in these cases. We also present conjectures for the Hilbert
series of $\mathcal E_{\tilde{D}_n}$, $\mathcal E_{\tilde{E}_6}$, and $\mathcal
E_{\tilde{E}_7}$, as well as for which graphs $G$ on six vertices $\mathcal
E_G$ is finite-dimensional.
