KMS states on the C∗-algebras of finite graphs

We consider a finite directed graph E, and the gauge action on its Toeplitz-Cuntz-Krieger algebra, viewed as an action of R. For inverse temperatures larger than a critical value βc, we give an explicit construction of all the KMSβ states. If the graph is strongly connected, then there is a unique KMSβc state, and this state factors through the quotient map onto C∗(E). Our approach is direct and relatively elementary.