Basics on positively multiplicative graphs and algebras

An oriented graph is said positively multiplicative when its adjacency matrix $A$ embeds in a matrix algebra admitting a basis $\mathsf{B}$ with nonnegative structure constants in which the matrix of the multiplication by $A$ coincides with $A$. The goal of this paper is to present basic properties of this notion and explain, through various simple examples, how it relates to highly non trivial problems like the combinatorial description of fusion rules, the description of the minimal boundary of graded graphs or the study of random walks on alcove tilings.