We study the generalised Bunce-Deddens algebras and their Toeplitz extensions
constructed by Kribs and Solel from a directed graph and a sequence ω of positive
integers. We describe both of these C*-algebras in terms of novel universal properties,
and prove uniqueness theorems for them; if ω determines an infinite supernatural
number, then no aperiodicity hypothesis is needed in our uniqueness theorem for the
generalised Bunce-Deddens algebra. We calculate the KMS states for the gauge action
in the Toeplitz algebra when the underlying graph is finite. We deduce that the generalised
Bunce-Deddens algebra is simple if and only if it supports exactly one KMS
state, and this is equivalent to the terms in the sequence ω all being coprime with the
period of the underlying graph.