This paper considers the notion of BL-algebras and converts any Galois finite field to a BL-algebra. It extends the class of BL-algebras based on finite fields and makes a connection between finite fields and BL-algebras. This study introduces the concept of a down digraph based on the notion of a down set (as a vertex) of BL-algebras, which states that any two vertices are adjacent if the related down sets are not joined. Characterization of the down digraphs is fundamental in this research, so we compute the number of edges in some down digraphs via combinators, to prove that down digraphs are connected under what conditions are connected and are Tournament. In the final, try to extract the BL-algebras from graphs, so show that any connected graph constructs a BL-algebra.