Digraph Groups Without Leaf Having An Arc Count One Greater Than Their Vertex

This paper investigates a particular class of digraph groups that are defined by non-empty balanced presentations. Each relation is expressed in the form R(x,y), where x and y are distinct generators, and R(⋅,⋅) is based on a fixed cyclically reduced word R(a,b) involving both a and b. A directed graph is constructed for each such presentation, where vertices correspond to generators and edges represent the relations. In previous research, Cihan identified 35 families of digraphs that satisfy |V(Γ)|=|A(Γ)|-1, of which 11 of them do not contain leaves. This paper demonstrates that, with two exceptions, the rank of the associated groups is either 1 or 2.