Some intrinsic properties of simple graph $C^*$-algebras
  To a directed graph $E$ is associated a $C^*$-algebra $C^* (E)$ called a graph $C^*$-algebra. There is a canonical action $\gamma$ of ${\bf T}$ on $C^* (E)$, called the gauge action. In this paper we present necessary and sufficient conditions for the fixed point algebra $C^* (E)^\gamma$ to be simple. Our results also yield some structure theorems for simple graph algebras. 
David Pask and Seung-Jai Rho