The co-universal C*-algebra of a row-finite graph

Let $E$ be a row-finite directed graph. We prove that there exists a $C^*$-algebra $\Cr{E}$ with the following co-universal property: given any $C^*$-algebra $B$ generated by a Toeplitz-Cuntz-Krieger $E$-family in which all the vertex projections are nonzero, there is a canonical homomorphism from $B$ onto $\Cr{E}$. We also identify when a homomorphism from $B$ to $\Cr{E}$ obtained from the co-universal property is injective. When every loop in $E$ has an entrance, $\Cr{E}$ coincides with the graph $C^*$-algebra $C^*(E)$, but in general, $\Cr{E}$ is a quotient of $C^*(E)$. We investigate the properties of $\Cr{E}$ with emphasis on the utility of co-universality as the defining property of the algebra.