Since the work of Cuntz and Krieger in 1980, there has been an interest in studying C*–algebras associated with subshifts and combinatorial objects, such as 0-1 matrices and graphs. Bates and Pask introduced a class of algebras associated with labelled graphs that generalised several of the previously mentioned algebras. A couple of decades before the work of Cuntz and Krieger, Leavitt studied a class of rings that do not satisfy the IBN. In the early 2000s, Ara et al. studied a purely algebraic analogue of the Cuntz-Krieger algebras, and they observed that when considering the matrix consisting only of 1’s (which in the C*-algebra are the Cuntz algebras), we obtain a subclass of Leavitt rings. Motivated by this work, Abrams and Pino introduced an algebraic analogue of C*-algebras of graphs, which they called Leavitt path algebras. These algebras share several properties and a way of proving this is by considering them as groupoid algebras/C*-algebras.

In this talk, I will present the C*-algebras of labelled graphs and their algebraic counterpart and I will explain how we can obtain groupoid models for these algebras. I will also talk about the approach using partial actions and give some conditions on partial actions of the free group such that they can be modelled using labelled graphs.