  We study invariance of KMS states on graph C*-algebras coming from strongly
connected and circulant graphs under the classical and quantum symmetry of the
graphs. We show that the unique KMS state for strongly connected graphs is
invariant under quantum automorphism group of the graph. For circulant graphs,
it is shown that the action of classical and quantum automorphism group
preserves only one of the KMS states occurring at the critical inverse
temperature. We also give an example of a graph C*-algebra having more than one
KMS state such that all of them are invariant under the action of classical
automorphism group of the graph, but there is a unique KMS state which is
invariant under the action of quantum automorphism group of the graph.
