  In recent joint work of the authors with Laca, we precisely formulated the
notion of partition function in the context of C*-dynamical systems. Here, we
compute the partition functions of C*-dynamical systems arising from Toeplitz
algebras of graphs, and we explicitly recover graph-theoretic information in
terms of C*-dynamical invariants. In addition, we compute the type for KMS
states on C*-algebras of finite (reducible) graphs and prove that the extremal
KMS states at critical inverse temperatures give rise to type III$_\lambda$
factors. Our starting point is an independent result parameterising the
partition functions of a certain class of C*-dynamical systems arising from
groupoid C*-algebras in terms of $\beta$-summable orbits.
