  The structure of KMS states of Toeplitz algebras associated to finite graphs
equipped with the gauge action is determined by an Huef--Laca--Raeburn--Sims.
Their results imply that extremal KMS states of type I correspond to vertices,
while extremal KMS states at critical inverse temperatures correspond to
minimal strongly connected components. The purpose of this article is to
clarify the role of non-minimal components and the relation between vertices
and minimal components in the KMS-structure. For each component $C_0$ and each
vertex $v \in C_0$, the KMS states at the critical inverse temperature of $v$
obtained by the limit of type I KMS states associated to $v$ uniquely
decomposes into a convex combination of KMS states associated to minimal
components. We show that for each minimal component $C$, the coefficient of the
KMS state associated to $C$ is nonzero if and only if there exists a maximal
path from $C$ to $C_0$ in the graph of components.
