Universal labeling algebras as invariants of layered graphs

In this work we will study the universal labeling algebra A(Γ)A(Γ), a related algebra B(Γ)B(Γ), and their behavior as invariants of layered graphs. We will introduce the notion of an upper vertex-like basis, which allows us to recover structural information about the graph Γ   from the algebra B(Γ)B(Γ). We will use these bases to show that several classes of layered graphs are uniquely identified by their corresponding algebras B(Γ)B(Γ).