Stability of $\boldsymbol{C^*}$-algebras associated to graphs

We characterize stability of graph $C^*$-algebras by giving five conditions equivalent to their stability. We also show that if $G$ is a graph with no sources, then $C^*(G)$ is stable if and only if each vertex in $G$ can be reached by an infinite number of vertices. We use this characterization to realize the stabilization of a graph $C^*$-algebra. Specifically, if $G$ is a graph and $\tilde{G}$ is the graph formed by adding a head to each vertex of $G$, then $C^*(\tilde{G})$ is the stabilization of $C^*(G)$; that is, $C^*(\tilde{G}) \cong C^*(G) \otimes \mathcal{K}$.