KMS states on the -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 β c , we give an explicit construction of all the KMS β states. If the graph is strongly connected, then there is a unique KMS β c state, and this state factors through the quotient map onto C ∗ ( E ) . Our approach is direct and relatively elementary.